{"id":409,"date":"2021-03-25T16:48:25","date_gmt":"2021-03-25T16:48:25","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus2\/?post_type=chapter&p=409"},"modified":"2021-11-17T23:53:21","modified_gmt":"2021-11-17T23:53:21","slug":"area-and-arc-length-in-polar-coordinates-2","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/area-and-arc-length-in-polar-coordinates-2\/","title":{"raw":"Problem Set: Area and Arc Length in Polar Coordinates","rendered":"Problem Set: Area and Arc Length in Polar Coordinates"},"content":{"raw":"

For the following exercises, determine a definite integral that represents the area.<\/p>\r\n\r\n

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1.\u00a0<\/strong>Region enclosed by [latex]r=4[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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2.\u00a0<\/strong>Region enclosed by [latex]r=3\\sin\\theta [\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"625363\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"625363\"][latex]\\frac{9}{2}{\\displaystyle\\int }_{0}^{\\pi }{\\sin}^{2}\\theta d\\theta [\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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3.\u00a0<\/strong>Region in the first quadrant within the cardioid [latex]r=1+\\sin\\theta [\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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4.\u00a0<\/strong>Region enclosed by one petal of [latex]r=8\\sin\\left(2\\theta \\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"145065\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"145065\"][latex]32{\\displaystyle\\int }_{0}^{\\frac{\\pi}{2}}{\\sin}^{2}\\left(2\\theta \\right)d\\theta [\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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5. <\/strong>Region enclosed by one petal of [latex]r=\\cos\\left(3\\theta \\right)[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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6.\u00a0<\/strong>Region below the polar axis and enclosed by [latex]r=1-\\sin\\theta [\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"625106\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"625106\"][latex]\\frac{1}{2}{\\displaystyle\\int }_{\\pi }^{2\\pi }{\\left(1-\\sin\\theta \\right)}^{2}d\\theta [\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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7.\u00a0<\/strong>Region in the first quadrant enclosed by [latex]r=2-\\cos\\theta [\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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8.\u00a0<\/strong>Region enclosed by the inner loop of [latex]r=2 - 3\\sin\\theta [\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"827654\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"827654\"][latex]{\\displaystyle\\int }_{{\\sin}^{-1}\\left(\\frac{2}{3}\\right)}^{\\frac{\\pi}{2}}{\\left(2 - 3\\sin\\theta \\right)}^{2}d\\theta [\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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9.\u00a0<\/strong>Region enclosed by the inner loop of [latex]r=3 - 4\\cos\\theta [\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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10. <\/strong>Region enclosed by [latex]r=1 - 2\\cos\\theta [\/latex] and outside the inner loop<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"755872\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"755872\"][latex]{\\displaystyle\\int }_{0}^{\\pi }{\\left(1 - 2\\cos\\theta \\right)}^{2}d\\theta -{\\displaystyle\\int }_{0}^{\\frac{\\pi}{3}}{\\left(1 - 2\\cos\\theta \\right)}^{2}d\\theta [\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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11.\u00a0<\/strong>Region common to [latex]r=3\\sin\\theta \\text{ and }r=2-\\sin\\theta [\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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12.\u00a0<\/strong>Region common to [latex]r=2\\text{ and }r=4\\cos\\theta [\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"630964\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"630964\"][latex]4{\\displaystyle\\int }_{0}^{\\frac{\\pi}{3}}d\\theta +16{\\displaystyle\\int }_{\\frac{\\pi}{3}}^{\\frac{\\pi}{2}}\\left({\\cos}^{2}\\theta \\right)d\\theta [\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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13.\u00a0<\/strong>Region common to [latex]r=3\\cos\\theta \\text{ and }r=3\\sin\\theta [\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n

For the following exercises, find the area of the described region.<\/p>\r\n\r\n

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14.\u00a0<\/strong>Enclosed by [latex]r=6\\sin\\theta [\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"876234\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"876234\"][latex]9\\pi [\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

15.\u00a0<\/strong>Above the polar axis enclosed by [latex]r=2+\\sin\\theta [\/latex]<\/span><\/div>\r\n<\/div>\r\n<\/div>\r\n
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16.\u00a0<\/strong>Below the polar axis and enclosed by [latex]r=2-\\cos\\theta [\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"334706\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"334706\"][latex]\\frac{9\\pi }{4}[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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17.\u00a0<\/strong>Enclosed by one petal of [latex]r=4\\cos\\left(3\\theta \\right)[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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18.\u00a0<\/strong>Enclosed by one petal of [latex]r=3\\cos\\left(2\\theta \\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"997340\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"997340\"][latex]\\frac{9\\pi }{8}[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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19.\u00a0<\/strong>Enclosed by [latex]r=1+\\sin\\theta [\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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20.\u00a0<\/strong>Enclosed by the inner loop of [latex]r=3+6\\cos\\theta [\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"374043\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"374043\"][latex]\\frac{18\\pi -27\\sqrt{3}}{2}[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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21.\u00a0<\/strong>Enclosed by [latex]r=2+4\\cos\\theta [\/latex] and outside the inner loop<\/div>\r\n<\/div>\r\n<\/div>\r\n
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22.\u00a0<\/strong>Common interior of [latex]r=4\\sin\\left(2\\theta \\right)\\text{and }r=2[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"693485\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"693485\"][latex]\\frac{4}{3}\\left(4\\pi -3\\sqrt{3}\\right)[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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23.\u00a0<\/strong>Common interior of [latex]r=3 - 2\\sin\\theta \\text{ and }r=-3+2\\sin\\theta [\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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24. <\/strong>Common interior of [latex]r=6\\sin\\theta \\text{ and }r=3[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"885115\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"885115\"][latex]\\frac{3}{2}\\left(4\\pi -3\\sqrt{3}\\right)[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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25.\u00a0<\/strong>Inside [latex]r=1+\\cos\\theta [\/latex] and outside [latex]r=\\cos\\theta [\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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26.\u00a0<\/strong>Common interior of [latex]r=2+2\\cos\\theta \\text{ and }r=2\\sin\\theta [\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"158557\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"158557\"][latex]2\\pi -4[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

For the following exercises, find a definite integral that represents the arc length.<\/p>\r\n\r\n

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27.\u00a0<\/strong>[latex]r=4\\cos\\theta \\text{ on the interval }0\\le \\theta \\le \\frac{\\pi }{2}[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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28.\u00a0<\/strong>[latex]r=1+\\sin\\theta [\/latex] on the interval [latex]0\\le \\theta \\le 2\\pi [\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"775158\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"775158\"][latex]{\\displaystyle\\int }_{0}^{2\\pi }\\sqrt{{\\left(1+\\sin\\theta \\right)}^{2}+{\\cos}^{2}\\theta }d\\theta [\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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29.\u00a0<\/strong>[latex]r=2\\sec\\theta \\text{ on the interval }0\\le \\theta \\le \\frac{\\pi }{3}[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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30.\u00a0<\/strong>[latex]r={e}^{\\theta }\\text{ on the interval }0\\le \\theta \\le 1[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"8149\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"8149\"][latex]\\sqrt{2}{\\displaystyle\\int }_{0}^{1}{e}^{\\theta }d\\theta [\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

For the following exercises, find the length of the curve over the given interval.<\/p>\r\n\r\n

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31.\u00a0<\/strong>[latex]r=6\\text{ on the interval }0\\le \\theta \\le \\frac{\\pi }{2}[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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32.\u00a0<\/strong>[latex]r={e}^{3\\theta }\\text{ on the interval }0\\le \\theta \\le 2[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"799417\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"799417\"][latex]\\frac{\\sqrt{10}}{3}\\left({e}^{6}-1\\right)[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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33.\u00a0<\/strong>[latex]r=6\\cos\\theta \\text{ on the interval }0\\le \\theta \\le \\frac{\\pi }{2}[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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34.\u00a0<\/strong>[latex]r=8+8\\cos\\theta \\text{ on the interval }0\\le \\theta \\le \\pi [\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"221912\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"221912\"]32[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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35.\u00a0<\/strong>[latex]r=1-\\sin\\theta \\text{ on the interval }0\\le \\theta \\le 2\\pi [\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n

For the following exercises, use the integration capabilities of a calculator to approximate the length of the curve.<\/p>\r\n\r\n

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36. [T]<\/strong> [latex]r=3\\theta \\text{ on the interval }0\\le \\theta \\le \\frac{\\pi }{2}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"474573\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"474573\"]6.238[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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37. [T]<\/strong> [latex]r=\\frac{2}{\\theta }\\text{ on the interval }\\pi \\le \\theta \\le 2\\pi [\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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38. [T]<\/strong> [latex]r={\\sin}^{2}\\left(\\frac{\\theta }{2}\\right)\\text{ on the interval }0\\le \\theta \\le \\pi [\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"109360\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"109360\"]2[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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39. [T]<\/strong> [latex]r=2{\\theta }^{2}\\text{ on the interval }0\\le \\theta \\le \\pi [\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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40. [T]<\/strong> [latex]r=\\sin\\left(3\\cos\\theta \\right)\\text{ on the interval }0\\le \\theta \\le \\pi [\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"746652\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"746652\"]4.39[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

For the following exercises, use the familiar formula from geometry to find the area of the region described and then confirm by using the definite integral.<\/span><\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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41.\u00a0<\/strong>[latex]r=3\\sin\\theta \\text{ on the interval }0\\le \\theta \\le \\pi [\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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42.\u00a0<\/strong>[latex]r=\\sin\\theta +\\cos\\theta \\text{ on the interval }0\\le \\theta \\le \\pi [\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"288047\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"288047\"][latex]A=\\pi {\\left(\\frac{\\sqrt{2}}{2}\\right)}^{2}=\\frac{\\pi }{2}\\text{ and }\\frac{1}{2}{\\displaystyle\\int }_{0}^{\\pi }\\left(1+2\\sin\\theta \\cos\\theta \\right)d\\theta =\\frac{\\pi }{2}[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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43.\u00a0<\/strong>[latex]r=6\\sin\\theta +8\\cos\\theta \\text{ on the interval }0\\le \\theta \\le \\pi [\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n

For the following exercises, use the familiar formula from geometry to find the length of the curve and then confirm using the definite integral.<\/p>\r\n\r\n

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44.\u00a0<\/strong>[latex]r=3\\sin\\theta \\text{ on the interval }0\\le \\theta \\le \\pi [\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"668754\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"668754\"][latex]C=2\\pi \\left(\\frac{3}{2}\\right)=3\\pi \\text{ and }{\\displaystyle\\int }_{0}^{\\pi }3d\\theta =3\\pi [\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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45.\u00a0<\/strong>[latex]r=\\sin\\theta +\\cos\\theta \\text{ on the interval }0\\le \\theta \\le \\pi [\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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46.\u00a0<\/strong>[latex]r=6\\sin\\theta +8\\cos\\theta \\text{ on the interval }0\\le \\theta \\le \\pi [\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"636380\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"636380\"][latex]C=2\\pi \\left(5\\right)=10\\pi \\text{ and }{\\displaystyle\\int }_{0}^{\\pi }10d\\theta =10\\pi [\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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47.\u00a0<\/strong>Verify that if [latex]y=r\\sin\\theta =f\\left(\\theta \\right)\\sin\\theta [\/latex] then [latex]\\frac{dy}{d\\theta }=f\\prime \\left(\\theta \\right)\\sin\\theta +f\\left(\\theta \\right)\\cos\\theta [\/latex].<\/div>\r\n<\/div>\r\n<\/div>\r\n

For the following exercises, find the slope of a tangent line to a polar curve [latex]r=f\\left(\\theta \\right)[\/latex]. Let [latex]x=r\\cos\\theta =f\\left(\\theta \\right)\\cos\\theta [\/latex] and [latex]y=r\\sin\\theta =f\\left(\\theta \\right)\\sin\\theta [\/latex], so the polar equation [latex]r=f\\left(\\theta \\right)[\/latex] is now written in parametric form.<\/p>\r\n\r\n

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48.\u00a0<\/strong>Use the definition of the derivative [latex]\\frac{dy}{dx}=\\frac{\\frac{dy}{d}\\theta }{\\frac{dx}{d}\\theta }[\/latex] and the product rule to derive the derivative of a polar equation.<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"944660\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"944660\"][latex]\\frac{dy}{dx}=\\frac{{f}^{\\prime }\\left(\\theta \\right)\\sin\\theta +f\\left(\\theta \\right)\\cos\\theta }{{f}^{\\prime }\\left(\\theta \\right)\\cos\\theta -f\\left(\\theta \\right)\\sin\\theta }[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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49.\u00a0<\/strong>[latex]r=1-\\sin\\theta [\/latex]; [latex]\\left(\\frac{1}{2},\\frac{\\pi }{6}\\right)[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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50.\u00a0<\/strong>[latex]r=4\\cos\\theta [\/latex]; [latex]\\left(2,\\frac{\\pi }{3}\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"436453\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"436453\"]The slope is [latex]\\frac{1}{\\sqrt{3}}[\/latex].[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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51.\u00a0<\/strong>[latex]r=8\\sin\\theta [\/latex]; [latex]\\left(4,\\frac{5\\pi }{6}\\right)[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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52.\u00a0<\/strong>[latex]r=4+\\sin\\theta [\/latex]; [latex]\\left(3,\\frac{3\\pi }{2}\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"127518\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"127518\"]The slope is 0.[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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53.\u00a0<\/strong>[latex]r=6+3\\cos\\theta [\/latex]; [latex]\\left(3,\\pi \\right)[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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54.\u00a0<\/strong>[latex]r=4\\cos\\left(2\\theta \\right)[\/latex]; tips of the leaves<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"832857\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"832857\"]At [latex]\\left(4,0\\right)[\/latex], the slope is undefined. At [latex]\\left(-4,\\frac{\\pi }{2}\\right)[\/latex], the slope is 0.[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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55.\u00a0<\/strong>[latex]r=2\\sin\\left(3\\theta \\right)[\/latex]; tips of the leaves<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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56.\u00a0<\/strong>[latex]r=2\\theta [\/latex]; [latex]\\left(\\frac{\\pi }{2},\\frac{\\pi }{4}\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"845887\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"845887\"]The slope is undefined at [latex]\\theta =\\frac{\\pi }{4}[\/latex].[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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57.\u00a0<\/strong>Find the points on the interval [latex]\\text{-}\\pi \\le \\theta \\le \\pi [\/latex] at which the cardioid [latex]r=1-\\cos\\theta [\/latex] has a vertical or horizontal tangent line.<\/div>\r\n<\/div>\r\n<\/div>\r\n
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58.\u00a0<\/strong>For the cardioid [latex]r=1+\\sin\\theta [\/latex], find the slope of the tangent line when [latex]\\theta =\\frac{\\pi }{3}[\/latex].<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"786285\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"786285\"]Slope = \u22121.[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

For the following exercises, find the slope of the tangent line to the given polar curve at the point given by the value of [latex]\\theta [\/latex].<\/p>\r\n\r\n

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59.\u00a0<\/strong>[latex]r=3\\cos\\theta ,\\theta =\\frac{\\pi }{3}[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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60.\u00a0<\/strong>[latex]r=\\theta [\/latex], [latex]\\theta =\\frac{\\pi }{2}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"495314\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"495314\"]Slope is [latex]\\frac{-2}{\\pi }[\/latex].[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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61.\u00a0<\/strong>[latex]r=\\text{ln}\\theta [\/latex], [latex]\\theta =e[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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62. [T]<\/strong> Use technology: [latex]r=2+4\\cos\\theta [\/latex] at [latex]\\theta =\\frac{\\pi }{6}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"55006\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"55006\"]Calculator answer: \u22120.836.[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

For the following exercises, find the points at which the following polar curves have a horizontal or vertical tangent line.<\/p>\r\n\r\n

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63.\u00a0<\/strong>[latex]r=4\\cos\\theta [\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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64.\u00a0<\/strong>[latex]{r}^{2}=4\\cos\\left(2\\theta \\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"429008\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"429008\"]Horizontal tangent at [latex]\\left(\\pm\\sqrt{2},\\frac{\\pi }{6}\\right)[\/latex], [latex]\\left(\\pm\\sqrt{2},-\\frac{\\pi }{6}\\right)[\/latex].[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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65.\u00a0<\/strong>[latex]r=2\\sin\\left(2\\theta \\right)[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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66.\u00a0<\/strong>The cardioid [latex]r=1+\\sin\\theta [\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"768582\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"768582\"]Horizontal tangents at [latex]\\frac{\\pi }{2},\\frac{7\\pi }{6},\\frac{11\\pi }{6}[\/latex]. Vertical tangents at [latex]\\frac{\\pi }{6},\\frac{5\\pi }{6}[\/latex] and also at the pole [latex]\\left(0,0\\right)[\/latex].[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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67.\u00a0<\/strong>Show that the curve [latex]r=\\sin\\theta \\tan\\theta [\/latex] (called a cissoid of Diocles<\/em><\/span>) has the line [latex]x=1[\/latex] as a vertical asymptote.<\/div>\r\n<\/div>\r\n<\/div>","rendered":"

For the following exercises, determine a definite integral that represents the area.<\/p>\n

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1.\u00a0<\/strong>Region enclosed by [latex]r=4[\/latex]<\/div>\n<\/div>\n<\/div>\n
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2.\u00a0<\/strong>Region enclosed by [latex]r=3\\sin\\theta[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]\\frac{9}{2}{\\displaystyle\\int }_{0}^{\\pi }{\\sin}^{2}\\theta d\\theta[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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3.\u00a0<\/strong>Region in the first quadrant within the cardioid [latex]r=1+\\sin\\theta[\/latex]<\/div>\n<\/div>\n<\/div>\n
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4.\u00a0<\/strong>Region enclosed by one petal of [latex]r=8\\sin\\left(2\\theta \\right)[\/latex]<\/p>\n<\/div>\n

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[latex]32{\\displaystyle\\int }_{0}^{\\frac{\\pi}{2}}{\\sin}^{2}\\left(2\\theta \\right)d\\theta[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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5. <\/strong>Region enclosed by one petal of [latex]r=\\cos\\left(3\\theta \\right)[\/latex]<\/div>\n<\/div>\n<\/div>\n
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6.\u00a0<\/strong>Region below the polar axis and enclosed by [latex]r=1-\\sin\\theta[\/latex]<\/p>\n<\/div>\n

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[latex]\\frac{1}{2}{\\displaystyle\\int }_{\\pi }^{2\\pi }{\\left(1-\\sin\\theta \\right)}^{2}d\\theta[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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7.\u00a0<\/strong>Region in the first quadrant enclosed by [latex]r=2-\\cos\\theta[\/latex]<\/div>\n<\/div>\n<\/div>\n
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8.\u00a0<\/strong>Region enclosed by the inner loop of [latex]r=2 - 3\\sin\\theta[\/latex]<\/p>\n<\/div>\n

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[latex]{\\displaystyle\\int }_{{\\sin}^{-1}\\left(\\frac{2}{3}\\right)}^{\\frac{\\pi}{2}}{\\left(2 - 3\\sin\\theta \\right)}^{2}d\\theta[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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9.\u00a0<\/strong>Region enclosed by the inner loop of [latex]r=3 - 4\\cos\\theta[\/latex]<\/div>\n<\/div>\n<\/div>\n
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10. <\/strong>Region enclosed by [latex]r=1 - 2\\cos\\theta[\/latex] and outside the inner loop<\/p>\n<\/div>\n

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[latex]{\\displaystyle\\int }_{0}^{\\pi }{\\left(1 - 2\\cos\\theta \\right)}^{2}d\\theta -{\\displaystyle\\int }_{0}^{\\frac{\\pi}{3}}{\\left(1 - 2\\cos\\theta \\right)}^{2}d\\theta[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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11.\u00a0<\/strong>Region common to [latex]r=3\\sin\\theta \\text{ and }r=2-\\sin\\theta[\/latex]<\/div>\n<\/div>\n<\/div>\n
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12.\u00a0<\/strong>Region common to [latex]r=2\\text{ and }r=4\\cos\\theta[\/latex]<\/p>\n<\/div>\n

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[latex]4{\\displaystyle\\int }_{0}^{\\frac{\\pi}{3}}d\\theta +16{\\displaystyle\\int }_{\\frac{\\pi}{3}}^{\\frac{\\pi}{2}}\\left({\\cos}^{2}\\theta \\right)d\\theta[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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13.\u00a0<\/strong>Region common to [latex]r=3\\cos\\theta \\text{ and }r=3\\sin\\theta[\/latex]<\/div>\n<\/div>\n<\/div>\n

For the following exercises, find the area of the described region.<\/p>\n

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14.\u00a0<\/strong>Enclosed by [latex]r=6\\sin\\theta[\/latex]<\/p>\n<\/div>\n

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[latex]9\\pi[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
15.\u00a0<\/strong>Above the polar axis enclosed by [latex]r=2+\\sin\\theta[\/latex]<\/span><\/div>\n<\/div>\n<\/div>\n
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16.\u00a0<\/strong>Below the polar axis and enclosed by [latex]r=2-\\cos\\theta[\/latex]<\/p>\n<\/div>\n

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[latex]\\frac{9\\pi }{4}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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17.\u00a0<\/strong>Enclosed by one petal of [latex]r=4\\cos\\left(3\\theta \\right)[\/latex]<\/div>\n<\/div>\n<\/div>\n
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18.\u00a0<\/strong>Enclosed by one petal of [latex]r=3\\cos\\left(2\\theta \\right)[\/latex]<\/p>\n<\/div>\n

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[latex]\\frac{9\\pi }{8}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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19.\u00a0<\/strong>Enclosed by [latex]r=1+\\sin\\theta[\/latex]<\/div>\n<\/div>\n<\/div>\n
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20.\u00a0<\/strong>Enclosed by the inner loop of [latex]r=3+6\\cos\\theta[\/latex]<\/p>\n<\/div>\n

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[latex]\\frac{18\\pi -27\\sqrt{3}}{2}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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21.\u00a0<\/strong>Enclosed by [latex]r=2+4\\cos\\theta[\/latex] and outside the inner loop<\/div>\n<\/div>\n<\/div>\n
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22.\u00a0<\/strong>Common interior of [latex]r=4\\sin\\left(2\\theta \\right)\\text{and }r=2[\/latex]<\/p>\n<\/div>\n

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[latex]\\frac{4}{3}\\left(4\\pi -3\\sqrt{3}\\right)[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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23.\u00a0<\/strong>Common interior of [latex]r=3 - 2\\sin\\theta \\text{ and }r=-3+2\\sin\\theta[\/latex]<\/div>\n<\/div>\n<\/div>\n
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24. <\/strong>Common interior of [latex]r=6\\sin\\theta \\text{ and }r=3[\/latex]<\/p>\n<\/div>\n

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[latex]\\frac{3}{2}\\left(4\\pi -3\\sqrt{3}\\right)[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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25.\u00a0<\/strong>Inside [latex]r=1+\\cos\\theta[\/latex] and outside [latex]r=\\cos\\theta[\/latex]<\/div>\n<\/div>\n<\/div>\n
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26.\u00a0<\/strong>Common interior of [latex]r=2+2\\cos\\theta \\text{ and }r=2\\sin\\theta[\/latex]<\/p>\n<\/div>\n

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[latex]2\\pi -4[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

For the following exercises, find a definite integral that represents the arc length.<\/p>\n

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27.\u00a0<\/strong>[latex]r=4\\cos\\theta \\text{ on the interval }0\\le \\theta \\le \\frac{\\pi }{2}[\/latex]<\/div>\n<\/div>\n<\/div>\n
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28.\u00a0<\/strong>[latex]r=1+\\sin\\theta[\/latex] on the interval [latex]0\\le \\theta \\le 2\\pi[\/latex]<\/p>\n<\/div>\n

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[latex]{\\displaystyle\\int }_{0}^{2\\pi }\\sqrt{{\\left(1+\\sin\\theta \\right)}^{2}+{\\cos}^{2}\\theta }d\\theta[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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29.\u00a0<\/strong>[latex]r=2\\sec\\theta \\text{ on the interval }0\\le \\theta \\le \\frac{\\pi }{3}[\/latex]<\/div>\n<\/div>\n<\/div>\n
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30.\u00a0<\/strong>[latex]r={e}^{\\theta }\\text{ on the interval }0\\le \\theta \\le 1[\/latex]<\/p>\n<\/div>\n

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[latex]\\sqrt{2}{\\displaystyle\\int }_{0}^{1}{e}^{\\theta }d\\theta[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

For the following exercises, find the length of the curve over the given interval.<\/p>\n

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31.\u00a0<\/strong>[latex]r=6\\text{ on the interval }0\\le \\theta \\le \\frac{\\pi }{2}[\/latex]<\/div>\n<\/div>\n<\/div>\n
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32.\u00a0<\/strong>[latex]r={e}^{3\\theta }\\text{ on the interval }0\\le \\theta \\le 2[\/latex]<\/p>\n<\/div>\n

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[latex]\\frac{\\sqrt{10}}{3}\\left({e}^{6}-1\\right)[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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33.\u00a0<\/strong>[latex]r=6\\cos\\theta \\text{ on the interval }0\\le \\theta \\le \\frac{\\pi }{2}[\/latex]<\/div>\n<\/div>\n<\/div>\n
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34.\u00a0<\/strong>[latex]r=8+8\\cos\\theta \\text{ on the interval }0\\le \\theta \\le \\pi[\/latex]<\/p>\n<\/div>\n

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32<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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35.\u00a0<\/strong>[latex]r=1-\\sin\\theta \\text{ on the interval }0\\le \\theta \\le 2\\pi[\/latex]<\/div>\n<\/div>\n<\/div>\n

For the following exercises, use the integration capabilities of a calculator to approximate the length of the curve.<\/p>\n

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36. [T]<\/strong> [latex]r=3\\theta \\text{ on the interval }0\\le \\theta \\le \\frac{\\pi }{2}[\/latex]<\/p>\n<\/div>\n

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6.238<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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37. [T]<\/strong> [latex]r=\\frac{2}{\\theta }\\text{ on the interval }\\pi \\le \\theta \\le 2\\pi[\/latex]<\/div>\n<\/div>\n<\/div>\n
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38. [T]<\/strong> [latex]r={\\sin}^{2}\\left(\\frac{\\theta }{2}\\right)\\text{ on the interval }0\\le \\theta \\le \\pi[\/latex]<\/p>\n<\/div>\n

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2<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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39. [T]<\/strong> [latex]r=2{\\theta }^{2}\\text{ on the interval }0\\le \\theta \\le \\pi[\/latex]<\/div>\n<\/div>\n<\/div>\n
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40. [T]<\/strong> [latex]r=\\sin\\left(3\\cos\\theta \\right)\\text{ on the interval }0\\le \\theta \\le \\pi[\/latex]<\/p>\n<\/div>\n

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4.39<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

For the following exercises, use the familiar formula from geometry to find the area of the region described and then confirm by using the definite integral.<\/span><\/p>\n<\/div>\n<\/div>\n

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41.\u00a0<\/strong>[latex]r=3\\sin\\theta \\text{ on the interval }0\\le \\theta \\le \\pi[\/latex]<\/div>\n<\/div>\n<\/div>\n
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42.\u00a0<\/strong>[latex]r=\\sin\\theta +\\cos\\theta \\text{ on the interval }0\\le \\theta \\le \\pi[\/latex]<\/p>\n<\/div>\n

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[latex]A=\\pi {\\left(\\frac{\\sqrt{2}}{2}\\right)}^{2}=\\frac{\\pi }{2}\\text{ and }\\frac{1}{2}{\\displaystyle\\int }_{0}^{\\pi }\\left(1+2\\sin\\theta \\cos\\theta \\right)d\\theta =\\frac{\\pi }{2}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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43.\u00a0<\/strong>[latex]r=6\\sin\\theta +8\\cos\\theta \\text{ on the interval }0\\le \\theta \\le \\pi[\/latex]<\/div>\n<\/div>\n<\/div>\n

For the following exercises, use the familiar formula from geometry to find the length of the curve and then confirm using the definite integral.<\/p>\n

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44.\u00a0<\/strong>[latex]r=3\\sin\\theta \\text{ on the interval }0\\le \\theta \\le \\pi[\/latex]<\/p>\n<\/div>\n

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[latex]C=2\\pi \\left(\\frac{3}{2}\\right)=3\\pi \\text{ and }{\\displaystyle\\int }_{0}^{\\pi }3d\\theta =3\\pi[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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45.\u00a0<\/strong>[latex]r=\\sin\\theta +\\cos\\theta \\text{ on the interval }0\\le \\theta \\le \\pi[\/latex]<\/div>\n<\/div>\n<\/div>\n
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46.\u00a0<\/strong>[latex]r=6\\sin\\theta +8\\cos\\theta \\text{ on the interval }0\\le \\theta \\le \\pi[\/latex]<\/p>\n<\/div>\n

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[latex]C=2\\pi \\left(5\\right)=10\\pi \\text{ and }{\\displaystyle\\int }_{0}^{\\pi }10d\\theta =10\\pi[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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47.\u00a0<\/strong>Verify that if [latex]y=r\\sin\\theta =f\\left(\\theta \\right)\\sin\\theta[\/latex] then [latex]\\frac{dy}{d\\theta }=f\\prime \\left(\\theta \\right)\\sin\\theta +f\\left(\\theta \\right)\\cos\\theta[\/latex].<\/div>\n<\/div>\n<\/div>\n

For the following exercises, find the slope of a tangent line to a polar curve [latex]r=f\\left(\\theta \\right)[\/latex]. Let [latex]x=r\\cos\\theta =f\\left(\\theta \\right)\\cos\\theta[\/latex] and [latex]y=r\\sin\\theta =f\\left(\\theta \\right)\\sin\\theta[\/latex], so the polar equation [latex]r=f\\left(\\theta \\right)[\/latex] is now written in parametric form.<\/p>\n

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48.\u00a0<\/strong>Use the definition of the derivative [latex]\\frac{dy}{dx}=\\frac{\\frac{dy}{d}\\theta }{\\frac{dx}{d}\\theta }[\/latex] and the product rule to derive the derivative of a polar equation.<\/p>\n<\/div>\n

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[latex]\\frac{dy}{dx}=\\frac{{f}^{\\prime }\\left(\\theta \\right)\\sin\\theta +f\\left(\\theta \\right)\\cos\\theta }{{f}^{\\prime }\\left(\\theta \\right)\\cos\\theta -f\\left(\\theta \\right)\\sin\\theta }[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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49.\u00a0<\/strong>[latex]r=1-\\sin\\theta[\/latex]; [latex]\\left(\\frac{1}{2},\\frac{\\pi }{6}\\right)[\/latex]<\/div>\n<\/div>\n<\/div>\n
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50.\u00a0<\/strong>[latex]r=4\\cos\\theta[\/latex]; [latex]\\left(2,\\frac{\\pi }{3}\\right)[\/latex]<\/p>\n<\/div>\n

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The slope is [latex]\\frac{1}{\\sqrt{3}}[\/latex].<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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51.\u00a0<\/strong>[latex]r=8\\sin\\theta[\/latex]; [latex]\\left(4,\\frac{5\\pi }{6}\\right)[\/latex]<\/div>\n<\/div>\n<\/div>\n
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52.\u00a0<\/strong>[latex]r=4+\\sin\\theta[\/latex]; [latex]\\left(3,\\frac{3\\pi }{2}\\right)[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
The slope is 0.<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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53.\u00a0<\/strong>[latex]r=6+3\\cos\\theta[\/latex]; [latex]\\left(3,\\pi \\right)[\/latex]<\/div>\n<\/div>\n<\/div>\n
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54.\u00a0<\/strong>[latex]r=4\\cos\\left(2\\theta \\right)[\/latex]; tips of the leaves<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
At [latex]\\left(4,0\\right)[\/latex], the slope is undefined. At [latex]\\left(-4,\\frac{\\pi }{2}\\right)[\/latex], the slope is 0.<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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55.\u00a0<\/strong>[latex]r=2\\sin\\left(3\\theta \\right)[\/latex]; tips of the leaves<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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56.\u00a0<\/strong>[latex]r=2\\theta[\/latex]; [latex]\\left(\\frac{\\pi }{2},\\frac{\\pi }{4}\\right)[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
The slope is undefined at [latex]\\theta =\\frac{\\pi }{4}[\/latex].<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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57.\u00a0<\/strong>Find the points on the interval [latex]\\text{-}\\pi \\le \\theta \\le \\pi[\/latex] at which the cardioid [latex]r=1-\\cos\\theta[\/latex] has a vertical or horizontal tangent line.<\/div>\n<\/div>\n<\/div>\n
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58.\u00a0<\/strong>For the cardioid [latex]r=1+\\sin\\theta[\/latex], find the slope of the tangent line when [latex]\\theta =\\frac{\\pi }{3}[\/latex].<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
Slope = \u22121.<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

For the following exercises, find the slope of the tangent line to the given polar curve at the point given by the value of [latex]\\theta[\/latex].<\/p>\n

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59.\u00a0<\/strong>[latex]r=3\\cos\\theta ,\\theta =\\frac{\\pi }{3}[\/latex]<\/div>\n<\/div>\n<\/div>\n
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60.\u00a0<\/strong>[latex]r=\\theta[\/latex], [latex]\\theta =\\frac{\\pi }{2}[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
Slope is [latex]\\frac{-2}{\\pi }[\/latex].<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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61.\u00a0<\/strong>[latex]r=\\text{ln}\\theta[\/latex], [latex]\\theta =e[\/latex]<\/div>\n<\/div>\n<\/div>\n
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62. [T]<\/strong> Use technology: [latex]r=2+4\\cos\\theta[\/latex] at [latex]\\theta =\\frac{\\pi }{6}[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
Calculator answer: \u22120.836.<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

For the following exercises, find the points at which the following polar curves have a horizontal or vertical tangent line.<\/p>\n

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63.\u00a0<\/strong>[latex]r=4\\cos\\theta[\/latex]<\/div>\n<\/div>\n<\/div>\n
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64.\u00a0<\/strong>[latex]{r}^{2}=4\\cos\\left(2\\theta \\right)[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
Horizontal tangent at [latex]\\left(\\pm\\sqrt{2},\\frac{\\pi }{6}\\right)[\/latex], [latex]\\left(\\pm\\sqrt{2},-\\frac{\\pi }{6}\\right)[\/latex].<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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65.\u00a0<\/strong>[latex]r=2\\sin\\left(2\\theta \\right)[\/latex]<\/div>\n<\/div>\n<\/div>\n
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66.\u00a0<\/strong>The cardioid [latex]r=1+\\sin\\theta[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
Horizontal tangents at [latex]\\frac{\\pi }{2},\\frac{7\\pi }{6},\\frac{11\\pi }{6}[\/latex]. Vertical tangents at [latex]\\frac{\\pi }{6},\\frac{5\\pi }{6}[\/latex] and also at the pole [latex]\\left(0,0\\right)[\/latex].<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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67.\u00a0<\/strong>Show that the curve [latex]r=\\sin\\theta \\tan\\theta[\/latex] (called a cissoid of Diocles<\/em><\/span>) has the line [latex]x=1[\/latex] as a vertical asymptote.<\/div>\n<\/div>\n<\/div>\n\n\t\t\t
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Candela Citations<\/h3>\n\t\t\t\t\t
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