{"id":83,"date":"2021-03-25T02:20:55","date_gmt":"2021-03-25T02:20:55","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/numerical-integration-3\/"},"modified":"2021-11-17T02:38:40","modified_gmt":"2021-11-17T02:38:40","slug":"numerical-integration-3","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/numerical-integration-3\/","title":{"raw":"Problem Set: Numerical Integration","rendered":"Problem Set: Numerical Integration"},"content":{"raw":"

Approximate the following integrals using either the midpoint rule, trapezoidal rule, or Simpson\u2019s rule as indicated. (Round answers to three decimal places.)<\/p>\r\n\r\n

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1.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{1}^{2}\\frac{dx}{x}[\/latex]; trapezoidal rule; [latex]n=5[\/latex]<\/p>\r\n[reveal-answer q=\"827592\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"827592\"]0.696[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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2.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{3}\\sqrt{4+{x}^{3}}dx[\/latex]; trapezoidal rule; [latex]n=6[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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3.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{3}\\sqrt{4+{x}^{3}}dx[\/latex]; Simpson\u2019s rule; [latex]n=3[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"418127\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"418127\"]9.298[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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4.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{12}{x}^{2}dx[\/latex]; midpoint rule; [latex]n=6[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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5.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{1}{\\sin}^{2}\\left(\\pi x\\right)dx[\/latex]; midpoint rule; [latex]n=3[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"127066\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"127066\"]0.5000[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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10.\u00a0<\/strong>Use the midpoint rule with eight subdivisions to estimate [latex]{\\displaystyle\\int }_{2}^{4}{x}^{2}dx[\/latex].<\/div>\r\n<\/div>\r\n<\/div>\r\n
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11.\u00a0<\/strong>Use the trapezoidal rule with four subdivisions to estimate [latex]{\\displaystyle\\int }_{2}^{4}{x}^{2}dx[\/latex].<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"351223\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"351223\"][latex]{T}_{4}=18.75[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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12.\u00a0<\/strong>Find the exact value of [latex]{\\displaystyle\\int }_{2}^{4}{x}^{2}dx[\/latex]. Find the error of approximation between the exact value and the value calculated using the trapezoidal rule with four subdivisions. Draw a graph to illustrate.<\/div>\r\n<\/div>\r\n<\/div>\r\n

Approximate the integral to three decimal places using the indicated rule.<\/p>\r\n\r\n

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13.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{1}{\\sin}^{2}\\left(\\pi x\\right)dx[\/latex]; trapezoidal rule; [latex]n=6[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"956341\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"956341\"]0.500[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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14.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{3}\\frac{1}{1+{x}^{3}}dx[\/latex]; trapezoidal rule; [latex]n=6[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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15.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{3}\\frac{1}{1+{x}^{3}}dx[\/latex]; Simpson\u2019s rule; [latex]n=3[\/latex]<\/p>\r\n[reveal-answer q=\"801833\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"801833\"]1.2819[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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16.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{0.8}{e}^{\\text{-}{x}^{2}}dx[\/latex]; trapezoidal rule; [latex]n=4[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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17.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{0.8}{e}^{\\text{-}{x}^{2}}dx[\/latex]; Simpson\u2019s rule; [latex]n=4[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"818159\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"818159\"]0.6577[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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18.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{0.4}\\sin\\left({x}^{2}\\right)dx[\/latex]; trapezoidal rule; [latex]n=4[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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19.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{0.4}\\sin\\left({x}^{2}\\right)dx[\/latex]; Simpson\u2019s rule; [latex]n=4[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"37168\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"37168\"]0.0213[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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20.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0.1}^{0.5}\\frac{\\cos{x}}{x}dx[\/latex]; trapezoidal rule; [latex]n=4[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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21.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0.1}^{0.5}\\frac{\\cos{x}}{x}dx[\/latex]; Simpson\u2019s rule; [latex]n=4[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"291916\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"291916\"]1.5629[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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22.\u00a0<\/strong>Evaluate [latex]{\\displaystyle\\int }_{0}^{1}\\frac{dx}{1+{x}^{2}}[\/latex] exactly and show that the result is [latex]\\frac{\\pi}{4}[\/latex]. Then, find the approximate value of the integral using the trapezoidal rule with [latex]n=4[\/latex] subdivisions. Use the result to approximate the value of [latex]\\pi [\/latex].<\/div>\r\n<\/div>\r\n<\/div>\r\n
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23.\u00a0<\/strong>Approximate [latex]{\\displaystyle\\int }_{2}^{4}\\frac{1}{\\text{ln}x}dx[\/latex] using the midpoint rule with four subdivisions to four decimal places.<\/p>\r\n[reveal-answer q=\"473622\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"473622\"]1.9133[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

24.\u00a0<\/strong>Approximate [latex]{\\displaystyle\\int }_{2}^{4}\\frac{1}{\\text{ln}x}dx[\/latex] using the trapezoidal rule with eight subdivisions to four decimal places.<\/span><\/div>\r\n<\/div>\r\n<\/div>\r\n
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25. <\/strong>Use the trapezoidal rule with four subdivisions to estimate [latex]{\\displaystyle\\int }_{0}^{0.8}{x}^{3}dx[\/latex] to four decimal places.<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"544276\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"544276\"][latex]\\text{T(4)}=0.1088[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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26.\u00a0<\/strong>Use the trapezoidal rule with four subdivisions to estimate [latex]{\\displaystyle\\int }_{0}^{0.8}{x}^{3}dx[\/latex]. Compare this value with the exact value and find the error estimate.<\/div>\r\n<\/div>\r\n<\/div>\r\n
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27.\u00a0<\/strong>Using Simpson\u2019s rule with four subdivisions, find [latex]{\\displaystyle\\int }_{0}^{\\frac{\\pi}{2}}\\cos\\left(x\\right)dx[\/latex].<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"939637\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"939637\"]1.0[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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28.\u00a0<\/strong>Show that the exact value of [latex]{\\displaystyle\\int }_{0}^{1}x{e}^{\\text{-}x}dx=1-\\frac{2}{e}[\/latex]. Find the absolute error if you approximate the integral using the midpoint rule with 16 subdivisions.<\/div>\r\n<\/div>\r\n<\/div>\r\n
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29.\u00a0<\/strong>Given [latex]{\\displaystyle\\int }_{0}^{1}x{e}^{\\text{-}x}dx=1-\\frac{2}{e}[\/latex], use the trapezoidal rule with 16 subdivisions to approximate the integral and find the absolute error.<\/p>\r\n[reveal-answer q=\"11024\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"11024\"]Approximate error is 0.000325.[\/hidden-answer]\r\n\r\n<\/div>\r\n

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30.\u00a0<\/strong>Find an upper bound for the error in estimating [latex]{\\displaystyle\\int }_{0}^{3}\\left(5x+4\\right)dx[\/latex] using the trapezoidal rule with six steps.<\/div>\r\n<\/div>\r\n<\/div>\r\n
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31.\u00a0<\/strong>Find an upper bound for the error in estimating [latex]{\\displaystyle\\int }_{4}^{5}\\frac{1}{{\\left(x - 1\\right)}^{2}}dx[\/latex] using the trapezoidal rule with seven subdivisions.<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"684822\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"684822\"][latex]\\frac{1}{7938}[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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32.\u00a0<\/strong>Find an upper bound for the error in estimating [latex]{\\displaystyle\\int }_{0}^{3}\\left(6{x}^{2}-1\\right)dx[\/latex] using Simpson\u2019s rule with [latex]n=10[\/latex] steps.<\/div>\r\n<\/div>\r\n<\/div>\r\n
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33.\u00a0<\/strong>Find an upper bound for the error in estimating [latex]{\\displaystyle\\int }_{2}^{5}\\frac{1}{x - 1}dx[\/latex] using Simpson\u2019s rule with [latex]n=10[\/latex] steps.<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"38378\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"38378\"][latex]\\frac{81}{25,000}[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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34.\u00a0<\/strong>Find an upper bound for the error in estimating [latex]{\\displaystyle\\int }_{0}^{\\pi }2x\\cos\\left(x\\right)dx[\/latex] using Simpson\u2019s rule with four steps.<\/div>\r\n<\/div>\r\n<\/div>\r\n
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35.\u00a0<\/strong>Estimate the minimum number of subintervals needed to approximate the integral [latex]{\\displaystyle\\int }_{1}^{4}\\left(5{x}^{2}+8\\right)dx[\/latex] with an error magnitude of less than 0.0001 using the trapezoidal rule.<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"177860\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"177860\"]475[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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36.\u00a0<\/strong>Determine a value of n<\/em> such that the trapezoidal rule will approximate [latex]{\\displaystyle\\int }_{0}^{1}\\sqrt{1+{x}^{2}}dx[\/latex] with an error of no more than 0.01.<\/div>\r\n<\/div>\r\n<\/div>\r\n
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37.\u00a0<\/strong>Estimate the minimum number of subintervals needed to approximate the integral [latex]{\\displaystyle\\int }_{2}^{3}\\left(2{x}^{3}+4x\\right)dx[\/latex] with an error of magnitude less than 0.0001 using the trapezoidal rule.<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"169963\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"169963\"]174[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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38.\u00a0<\/strong>Estimate the minimum number of subintervals needed to approximate the integral [latex]{\\displaystyle\\int }_{3}^{4}\\frac{1}{{\\left(x - 1\\right)}^{2}}dx[\/latex] with an error magnitude of less than 0.0001 using the trapezoidal rule.<\/div>\r\n<\/div>\r\n<\/div>\r\n
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39.\u00a0<\/strong>Use Simpson\u2019s rule with four subdivisions to approximate the area under the probability density function [latex]y=\\frac{1}{\\sqrt{2\\pi }}{e}^{\\frac{\\text{-}{x}^{2}}{2}}[\/latex] from [latex]x=0[\/latex] to [latex]x=0.4[\/latex].<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"199016\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"199016\"]0.1544[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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40.\u00a0<\/strong>Use Simpson\u2019s rule with [latex]n=14[\/latex] to approximate (to three decimal places) the area of the region bounded by the graphs of [latex]y=0[\/latex], [latex]x=0[\/latex], and [latex]x=\\frac{\\pi}{2}[\/latex].<\/div>\r\n<\/div>\r\n<\/div>\r\n
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41.\u00a0<\/strong>The length of one arch of the curve [latex]y=3\\sin\\left(2x\\right)[\/latex] is given by [latex]L={\\displaystyle\\int }_{0}^{\\frac{\\pi}{2}}\\sqrt{1+36{\\cos}^{2}\\left(2x\\right)}dx[\/latex]. Estimate L<\/em> using the trapezoidal rule with [latex]n=6[\/latex].<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"931225\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"931225\"]6.2807[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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42.\u00a0<\/strong>The length of the ellipse [latex]x=a\\cos\\left(t\\right),y=b\\sin\\left(t\\right),0\\le t\\le 2\\pi [\/latex] is given by [latex]L=4a{\\displaystyle\\int }_{0}^{\\frac{\\pi}{2}}\\sqrt{1-{e}^{2}{\\cos}^{2}\\left(t\\right)}dt[\/latex], where e<\/em> is the eccentricity of the ellipse. Use Simpson\u2019s rule with [latex]n=6[\/latex] subdivisions to estimate the length of the ellipse when [latex]a=2[\/latex] and [latex]e=\\frac{1}{3}[\/latex].<\/div>\r\n<\/div>\r\n<\/div>\r\n
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43.\u00a0<\/strong>Estimate the area of the surface generated by revolving the curve [latex]y=\\cos\\left(2x\\right),0\\le x\\le \\frac{\\pi }{4}[\/latex] about the x<\/em>-axis. Use the trapezoidal rule with six subdivisions.<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"458336\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"458336\"]4.606[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

44.\u00a0<\/strong>Estimate the area of the surface generated by revolving the curve [latex]y=2{x}^{2}[\/latex], [latex]0\\le x\\le 3[\/latex] about the <\/span>x-<\/em>axis. Use Simpson\u2019s rule with [latex]n=6[\/latex].<\/span><\/div>\r\n<\/div>\r\n<\/div>\r\n
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45.\u00a0<\/strong>The growth rate of a certain tree (in feet) is given by [latex]y=\\frac{2}{t+1}+{e}^{\\frac{\\text{-}{t}^{2}}{2}}[\/latex], where t<\/em> is time in years. Estimate the growth of the tree through the end of the second year by using Simpson\u2019s rule, using two subintervals. (Round the answer to the nearest hundredth.)<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"428992\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"428992\"]3.41 ft[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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46. [T]<\/strong> Use a calculator to approximate [latex]{\\displaystyle\\int }_{0}^{1}\\sin\\left(\\pi x\\right)dx[\/latex] using the midpoint rule with 25 subdivisions. Compute the relative error of approximation.<\/div>\r\n<\/div>\r\n<\/div>\r\n
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47. [T]<\/strong> Given [latex]{\\displaystyle\\int }_{1}^{5}\\left(3{x}^{2}-2x\\right)dx=100[\/latex], approximate the value of this integral using the trapezoidal rule with 16 subdivisions and determine the absolute error.<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"311781\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"311781\"][latex]{T}_{16}=100.125[\/latex]; absolute error = 0.125[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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48.\u00a0<\/strong>Given that we know the Fundamental Theorem of Calculus, why would we want to develop numerical methods for definite integrals?<\/div>\r\n<\/div>\r\n<\/div>\r\n
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49.\u00a0<\/strong>The table represents the coordinates [latex]\\left(x,\\text{ }y\\right)[\/latex] that give the boundary of a lot. The units of measurement are meters. Use the trapezoidal rule to estimate the number of square meters of land that is in this lot.<\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
x<\/em><\/th>\r\ny<\/em><\/th>\r\nx<\/em><\/th>\r\ny<\/em><\/th>\r\n<\/tr>\r\n<\/thead>\r\n
0<\/td>\r\n125<\/td>\r\n600<\/td>\r\n95<\/td>\r\n<\/tr>\r\n
100<\/td>\r\n125<\/td>\r\n700<\/td>\r\n88<\/td>\r\n<\/tr>\r\n
200<\/td>\r\n120<\/td>\r\n800<\/td>\r\n75<\/td>\r\n<\/tr>\r\n
300<\/td>\r\n112<\/td>\r\n900<\/td>\r\n35<\/td>\r\n<\/tr>\r\n
400<\/td>\r\n90<\/td>\r\n1000<\/td>\r\n0<\/td>\r\n<\/tr>\r\n
500<\/td>\r\n90<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n
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[reveal-answer q=\"747834\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"747834\"]about 89,250 [latex]\\text{m}^{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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50.\u00a0<\/strong>Choose the correct answer. When Simpson\u2019s rule is used to approximate the definite integral, it is necessary that the number of partitions be____<\/p>\r\n\r\n

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  1. an even number<\/li>\r\n \t
  2. odd number<\/li>\r\n \t
  3. either an even or an odd number<\/li>\r\n \t
  4. a multiple of 4<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n
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    51.\u00a0<\/strong>The \"Simpson\" sum is based on the area under a ____.<\/p>\r\n\r\n<\/div>\r\n

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    [reveal-answer q=\"233583\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"233583\"]parabola[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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    52.\u00a0<\/strong>The error formula for Simpson\u2019s rule depends on___.<\/p>\r\n\r\n

      \r\n \t
    1. [latex]f\\left(x\\right)[\/latex]<\/li>\r\n \t
    2. [latex]{f}^{\\prime }\\left(x\\right)[\/latex]<\/li>\r\n \t
    3. [latex]{f}^{\\left(4\\right)}\\left(x\\right)[\/latex]<\/li>\r\n \t
    4. the number of steps<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>","rendered":"

      Approximate the following integrals using either the midpoint rule, trapezoidal rule, or Simpson\u2019s rule as indicated. (Round answers to three decimal places.)<\/p>\n

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      1.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{1}^{2}\\frac{dx}{x}[\/latex]; trapezoidal rule; [latex]n=5[\/latex]<\/p>\n

      Show Solution<\/span><\/p>\n
      0.696<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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      2.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{3}\\sqrt{4+{x}^{3}}dx[\/latex]; trapezoidal rule; [latex]n=6[\/latex]<\/div>\n<\/div>\n<\/div>\n
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      3.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{3}\\sqrt{4+{x}^{3}}dx[\/latex]; Simpson\u2019s rule; [latex]n=3[\/latex]<\/p>\n<\/div>\n

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      Show Solution<\/span><\/p>\n
      9.298<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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      4.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{12}{x}^{2}dx[\/latex]; midpoint rule; [latex]n=6[\/latex]<\/div>\n<\/div>\n<\/div>\n
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      5.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{1}{\\sin}^{2}\\left(\\pi x\\right)dx[\/latex]; midpoint rule; [latex]n=3[\/latex]<\/p>\n<\/div>\n

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      Show Solution<\/span><\/p>\n
      0.5000<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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      10.\u00a0<\/strong>Use the midpoint rule with eight subdivisions to estimate [latex]{\\displaystyle\\int }_{2}^{4}{x}^{2}dx[\/latex].<\/div>\n<\/div>\n<\/div>\n
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      11.\u00a0<\/strong>Use the trapezoidal rule with four subdivisions to estimate [latex]{\\displaystyle\\int }_{2}^{4}{x}^{2}dx[\/latex].<\/p>\n<\/div>\n

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      Show Solution<\/span><\/p>\n
      [latex]{T}_{4}=18.75[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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      12.\u00a0<\/strong>Find the exact value of [latex]{\\displaystyle\\int }_{2}^{4}{x}^{2}dx[\/latex]. Find the error of approximation between the exact value and the value calculated using the trapezoidal rule with four subdivisions. Draw a graph to illustrate.<\/div>\n<\/div>\n<\/div>\n

      Approximate the integral to three decimal places using the indicated rule.<\/p>\n

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      13.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{1}{\\sin}^{2}\\left(\\pi x\\right)dx[\/latex]; trapezoidal rule; [latex]n=6[\/latex]<\/p>\n<\/div>\n

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      Show Solution<\/span><\/p>\n
      0.500<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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      14.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{3}\\frac{1}{1+{x}^{3}}dx[\/latex]; trapezoidal rule; [latex]n=6[\/latex]<\/div>\n<\/div>\n<\/div>\n
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      15.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{3}\\frac{1}{1+{x}^{3}}dx[\/latex]; Simpson\u2019s rule; [latex]n=3[\/latex]<\/p>\n

      Show Solution<\/span><\/p>\n
      1.2819<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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      16.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{0.8}{e}^{\\text{-}{x}^{2}}dx[\/latex]; trapezoidal rule; [latex]n=4[\/latex]<\/div>\n<\/div>\n<\/div>\n
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      17.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{0.8}{e}^{\\text{-}{x}^{2}}dx[\/latex]; Simpson\u2019s rule; [latex]n=4[\/latex]<\/p>\n<\/div>\n

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      Show Solution<\/span><\/p>\n
      0.6577<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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      18.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{0.4}\\sin\\left({x}^{2}\\right)dx[\/latex]; trapezoidal rule; [latex]n=4[\/latex]<\/div>\n<\/div>\n<\/div>\n
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      19.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{0.4}\\sin\\left({x}^{2}\\right)dx[\/latex]; Simpson\u2019s rule; [latex]n=4[\/latex]<\/p>\n<\/div>\n

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      Show Solution<\/span><\/p>\n
      0.0213<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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      20.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0.1}^{0.5}\\frac{\\cos{x}}{x}dx[\/latex]; trapezoidal rule; [latex]n=4[\/latex]<\/div>\n<\/div>\n<\/div>\n
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      21.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0.1}^{0.5}\\frac{\\cos{x}}{x}dx[\/latex]; Simpson\u2019s rule; [latex]n=4[\/latex]<\/p>\n<\/div>\n

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      Show Solution<\/span><\/p>\n
      1.5629<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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      22.\u00a0<\/strong>Evaluate [latex]{\\displaystyle\\int }_{0}^{1}\\frac{dx}{1+{x}^{2}}[\/latex] exactly and show that the result is [latex]\\frac{\\pi}{4}[\/latex]. Then, find the approximate value of the integral using the trapezoidal rule with [latex]n=4[\/latex] subdivisions. Use the result to approximate the value of [latex]\\pi[\/latex].<\/div>\n<\/div>\n<\/div>\n
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      23.\u00a0<\/strong>Approximate [latex]{\\displaystyle\\int }_{2}^{4}\\frac{1}{\\text{ln}x}dx[\/latex] using the midpoint rule with four subdivisions to four decimal places.<\/p>\n

      Show Solution<\/span><\/p>\n
      1.9133<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      24.\u00a0<\/strong>Approximate [latex]{\\displaystyle\\int }_{2}^{4}\\frac{1}{\\text{ln}x}dx[\/latex] using the trapezoidal rule with eight subdivisions to four decimal places.<\/span><\/div>\n<\/div>\n<\/div>\n
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      25. <\/strong>Use the trapezoidal rule with four subdivisions to estimate [latex]{\\displaystyle\\int }_{0}^{0.8}{x}^{3}dx[\/latex] to four decimal places.<\/p>\n<\/div>\n

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      Show Solution<\/span><\/p>\n
      [latex]\\text{T(4)}=0.1088[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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      26.\u00a0<\/strong>Use the trapezoidal rule with four subdivisions to estimate [latex]{\\displaystyle\\int }_{0}^{0.8}{x}^{3}dx[\/latex]. Compare this value with the exact value and find the error estimate.<\/div>\n<\/div>\n<\/div>\n
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      27.\u00a0<\/strong>Using Simpson\u2019s rule with four subdivisions, find [latex]{\\displaystyle\\int }_{0}^{\\frac{\\pi}{2}}\\cos\\left(x\\right)dx[\/latex].<\/p>\n<\/div>\n

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      Show Solution<\/span><\/p>\n
      1.0<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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      28.\u00a0<\/strong>Show that the exact value of [latex]{\\displaystyle\\int }_{0}^{1}x{e}^{\\text{-}x}dx=1-\\frac{2}{e}[\/latex]. Find the absolute error if you approximate the integral using the midpoint rule with 16 subdivisions.<\/div>\n<\/div>\n<\/div>\n
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      29.\u00a0<\/strong>Given [latex]{\\displaystyle\\int }_{0}^{1}x{e}^{\\text{-}x}dx=1-\\frac{2}{e}[\/latex], use the trapezoidal rule with 16 subdivisions to approximate the integral and find the absolute error.<\/p>\n

      Show Solution<\/span><\/p>\n
      Approximate error is 0.000325.<\/div>\n<\/div>\n<\/div>\n
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      30.\u00a0<\/strong>Find an upper bound for the error in estimating [latex]{\\displaystyle\\int }_{0}^{3}\\left(5x+4\\right)dx[\/latex] using the trapezoidal rule with six steps.<\/div>\n<\/div>\n<\/div>\n
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      31.\u00a0<\/strong>Find an upper bound for the error in estimating [latex]{\\displaystyle\\int }_{4}^{5}\\frac{1}{{\\left(x - 1\\right)}^{2}}dx[\/latex] using the trapezoidal rule with seven subdivisions.<\/p>\n<\/div>\n

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      Show Solution<\/span><\/p>\n
      [latex]\\frac{1}{7938}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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      32.\u00a0<\/strong>Find an upper bound for the error in estimating [latex]{\\displaystyle\\int }_{0}^{3}\\left(6{x}^{2}-1\\right)dx[\/latex] using Simpson\u2019s rule with [latex]n=10[\/latex] steps.<\/div>\n<\/div>\n<\/div>\n
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      33.\u00a0<\/strong>Find an upper bound for the error in estimating [latex]{\\displaystyle\\int }_{2}^{5}\\frac{1}{x - 1}dx[\/latex] using Simpson\u2019s rule with [latex]n=10[\/latex] steps.<\/p>\n<\/div>\n

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      Show Solution<\/span><\/p>\n
      [latex]\\frac{81}{25,000}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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      34.\u00a0<\/strong>Find an upper bound for the error in estimating [latex]{\\displaystyle\\int }_{0}^{\\pi }2x\\cos\\left(x\\right)dx[\/latex] using Simpson\u2019s rule with four steps.<\/div>\n<\/div>\n<\/div>\n
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      35.\u00a0<\/strong>Estimate the minimum number of subintervals needed to approximate the integral [latex]{\\displaystyle\\int }_{1}^{4}\\left(5{x}^{2}+8\\right)dx[\/latex] with an error magnitude of less than 0.0001 using the trapezoidal rule.<\/p>\n<\/div>\n

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      Show Solution<\/span><\/p>\n
      475<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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      36.\u00a0<\/strong>Determine a value of n<\/em> such that the trapezoidal rule will approximate [latex]{\\displaystyle\\int }_{0}^{1}\\sqrt{1+{x}^{2}}dx[\/latex] with an error of no more than 0.01.<\/div>\n<\/div>\n<\/div>\n
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      37.\u00a0<\/strong>Estimate the minimum number of subintervals needed to approximate the integral [latex]{\\displaystyle\\int }_{2}^{3}\\left(2{x}^{3}+4x\\right)dx[\/latex] with an error of magnitude less than 0.0001 using the trapezoidal rule.<\/p>\n<\/div>\n

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      Show Solution<\/span><\/p>\n
      174<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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      38.\u00a0<\/strong>Estimate the minimum number of subintervals needed to approximate the integral [latex]{\\displaystyle\\int }_{3}^{4}\\frac{1}{{\\left(x - 1\\right)}^{2}}dx[\/latex] with an error magnitude of less than 0.0001 using the trapezoidal rule.<\/div>\n<\/div>\n<\/div>\n
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      39.\u00a0<\/strong>Use Simpson\u2019s rule with four subdivisions to approximate the area under the probability density function [latex]y=\\frac{1}{\\sqrt{2\\pi }}{e}^{\\frac{\\text{-}{x}^{2}}{2}}[\/latex] from [latex]x=0[\/latex] to [latex]x=0.4[\/latex].<\/p>\n<\/div>\n

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      Show Solution<\/span><\/p>\n
      0.1544<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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      40.\u00a0<\/strong>Use Simpson\u2019s rule with [latex]n=14[\/latex] to approximate (to three decimal places) the area of the region bounded by the graphs of [latex]y=0[\/latex], [latex]x=0[\/latex], and [latex]x=\\frac{\\pi}{2}[\/latex].<\/div>\n<\/div>\n<\/div>\n
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      41.\u00a0<\/strong>The length of one arch of the curve [latex]y=3\\sin\\left(2x\\right)[\/latex] is given by [latex]L={\\displaystyle\\int }_{0}^{\\frac{\\pi}{2}}\\sqrt{1+36{\\cos}^{2}\\left(2x\\right)}dx[\/latex]. Estimate L<\/em> using the trapezoidal rule with [latex]n=6[\/latex].<\/p>\n<\/div>\n

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      Show Solution<\/span><\/p>\n
      6.2807<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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      42.\u00a0<\/strong>The length of the ellipse [latex]x=a\\cos\\left(t\\right),y=b\\sin\\left(t\\right),0\\le t\\le 2\\pi[\/latex] is given by [latex]L=4a{\\displaystyle\\int }_{0}^{\\frac{\\pi}{2}}\\sqrt{1-{e}^{2}{\\cos}^{2}\\left(t\\right)}dt[\/latex], where e<\/em> is the eccentricity of the ellipse. Use Simpson\u2019s rule with [latex]n=6[\/latex] subdivisions to estimate the length of the ellipse when [latex]a=2[\/latex] and [latex]e=\\frac{1}{3}[\/latex].<\/div>\n<\/div>\n<\/div>\n
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      43.\u00a0<\/strong>Estimate the area of the surface generated by revolving the curve [latex]y=\\cos\\left(2x\\right),0\\le x\\le \\frac{\\pi }{4}[\/latex] about the x<\/em>-axis. Use the trapezoidal rule with six subdivisions.<\/p>\n<\/div>\n

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      Show Solution<\/span><\/p>\n
      4.606<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      44.\u00a0<\/strong>Estimate the area of the surface generated by revolving the curve [latex]y=2{x}^{2}[\/latex], [latex]0\\le x\\le 3[\/latex] about the <\/span>x-<\/em>axis. Use Simpson\u2019s rule with [latex]n=6[\/latex].<\/span><\/div>\n<\/div>\n<\/div>\n
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      45.\u00a0<\/strong>The growth rate of a certain tree (in feet) is given by [latex]y=\\frac{2}{t+1}+{e}^{\\frac{\\text{-}{t}^{2}}{2}}[\/latex], where t<\/em> is time in years. Estimate the growth of the tree through the end of the second year by using Simpson\u2019s rule, using two subintervals. (Round the answer to the nearest hundredth.)<\/p>\n<\/div>\n

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      Show Solution<\/span><\/p>\n
      3.41 ft<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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      46. [T]<\/strong> Use a calculator to approximate [latex]{\\displaystyle\\int }_{0}^{1}\\sin\\left(\\pi x\\right)dx[\/latex] using the midpoint rule with 25 subdivisions. Compute the relative error of approximation.<\/div>\n<\/div>\n<\/div>\n
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      47. [T]<\/strong> Given [latex]{\\displaystyle\\int }_{1}^{5}\\left(3{x}^{2}-2x\\right)dx=100[\/latex], approximate the value of this integral using the trapezoidal rule with 16 subdivisions and determine the absolute error.<\/p>\n<\/div>\n

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      Show Solution<\/span><\/p>\n
      [latex]{T}_{16}=100.125[\/latex]; absolute error = 0.125<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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      48.\u00a0<\/strong>Given that we know the Fundamental Theorem of Calculus, why would we want to develop numerical methods for definite integrals?<\/div>\n<\/div>\n<\/div>\n
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      49.\u00a0<\/strong>The table represents the coordinates [latex]\\left(x,\\text{ }y\\right)[\/latex] that give the boundary of a lot. The units of measurement are meters. Use the trapezoidal rule to estimate the number of square meters of land that is in this lot.<\/p>\n\n\n\n\n\n\n\n\n\n\n
      x<\/em><\/th>\ny<\/em><\/th>\nx<\/em><\/th>\ny<\/em><\/th>\n<\/tr>\n<\/thead>\n
      0<\/td>\n125<\/td>\n600<\/td>\n95<\/td>\n<\/tr>\n
      100<\/td>\n125<\/td>\n700<\/td>\n88<\/td>\n<\/tr>\n
      200<\/td>\n120<\/td>\n800<\/td>\n75<\/td>\n<\/tr>\n
      300<\/td>\n112<\/td>\n900<\/td>\n35<\/td>\n<\/tr>\n
      400<\/td>\n90<\/td>\n1000<\/td>\n0<\/td>\n<\/tr>\n
      500<\/td>\n90<\/td>\n<\/td>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n
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      Show Solution<\/span><\/p>\n
      about 89,250 [latex]\\text{m}^{2}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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      50.\u00a0<\/strong>Choose the correct answer. When Simpson\u2019s rule is used to approximate the definite integral, it is necessary that the number of partitions be____<\/p>\n

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      1. an even number<\/li>\n
      2. odd number<\/li>\n
      3. either an even or an odd number<\/li>\n
      4. a multiple of 4<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n
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        51.\u00a0<\/strong>The “Simpson” sum is based on the area under a ____.<\/p>\n<\/div>\n

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        Show Solution<\/span><\/p>\n
        parabola<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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        52.\u00a0<\/strong>The error formula for Simpson\u2019s rule depends on___.<\/p>\n

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        1. [latex]f\\left(x\\right)[\/latex]<\/li>\n
        2. [latex]{f}^{\\prime }\\left(x\\right)[\/latex]<\/li>\n
        3. [latex]{f}^{\\left(4\\right)}\\left(x\\right)[\/latex]<\/li>\n
        4. the number of steps<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/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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