{"id":407,"date":"2021-03-25T16:48:08","date_gmt":"2021-03-25T16:48:08","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus2\/?post_type=chapter&p=407"},"modified":"2022-01-03T18:56:54","modified_gmt":"2022-01-03T18:56:54","slug":"calculus-of-parametric-curves-2","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/calculus-of-parametric-curves-2\/","title":{"raw":"Problem Set: Calculus of Parametric Curves","rendered":"Problem Set: Calculus of Parametric Curves"},"content":{"raw":"

For the following exercises, each set of parametric equations represents a line. Without eliminating the parameter, find the slope of each line.<\/p>\r\n\r\n

\r\n
\r\n
1.\u00a0<\/strong>[latex]\\begin{array}{cc}x=3+t,\\hfill & y=1-t\\hfill \\end{array}[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
\r\n
\r\n
\r\n
\r\n

2. <\/strong>[latex]\\begin{array}{cc}x=8+2t,\\hfill & y=1\\hfill \\end{array}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

\r\n

[reveal-answer q=\"611163\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"611163\"]0[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

\r\n
\r\n
3.\u00a0<\/strong>[latex]\\begin{array}{cc}x=4 - 3t,\\hfill & y=-2+6t\\hfill \\end{array}[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
\r\n
\r\n
\r\n
\r\n

4.\u00a0<\/strong>[latex]\\begin{array}{cc}x=-5t+7,\\hfill & y=3t - 1\\hfill \\end{array}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

\r\n

[reveal-answer q=\"988344\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"988344\"][latex]\\frac{-3}{5}[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

For the following exercises, determine the slope of the tangent line, then find the equation of the tangent line at the given value of the parameter.<\/p>\r\n\r\n

\r\n
\r\n
\r\n
\r\n
\r\n

5.\u00a0<\/strong>[latex]\\begin{array}{cc}x=3\\sin{t},\\hfill & y=3\\cos{t},t=\\frac{\\pi }{4}\\hfill \\end{array}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

\r\n
\r\n
\r\n
\r\n

6.\u00a0<\/strong>[latex]\\begin{array}{cc}x=\\cos{t},\\hfill & y=8\\sin{t},\\hfill \\end{array}t=\\frac{\\pi }{2}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

\r\n

[reveal-answer q=\"913282\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"913282\"][latex]\\text{Slope}=0[\/latex]; [latex]y=8[\/latex].[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

\r\n
\r\n
7.\u00a0<\/strong>[latex]\\begin{array}{cc}x=2t,\\hfill & y={t}^{3},t=-1\\hfill \\end{array}[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
\r\n
\r\n
\r\n
\r\n

8.\u00a0<\/strong>[latex]\\begin{array}{cc}x=t+\\frac{1}{t},\\hfill & y=t-\\frac{1}{t},t=1\\hfill \\end{array}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

\r\n

[reveal-answer q=\"176768\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"176768\"]Slope is undefined; [latex]x=2[\/latex].[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

\r\n
\r\n
\r\n
\r\n
\r\n

9.\u00a0<\/strong>[latex]\\begin{array}{cc}x=\\sqrt{t},\\hfill & y=2t,t=4\\hfill \\end{array}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

For the following exercises, find all points on the curve that have the given slope.<\/p>\r\n\r\n

\r\n
\r\n
\r\n
\r\n

10.\u00a0<\/strong>[latex]\\begin{array}{cc}x=4\\cos{t},\\hfill & y=4\\sin{t},\\hfill \\end{array}[\/latex] slope = 0.5<\/p>\r\n\r\n<\/div>\r\n

\r\n

[reveal-answer q=\"115186\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"115186\"][latex]\\tan{t}=\\left(-2\\right)[\/latex] [latex]\\left(\\frac{4}{\\sqrt{5}},\\frac{-8}{\\sqrt{5}}\\right),\\left(\\frac{4}{\\sqrt{5}},\\frac{-8}{\\sqrt{5}}\\right)[\/latex].[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

\r\n
\r\n
11.\u00a0<\/strong>[latex]\\begin{array}{cc}x=2\\cos{t},\\hfill & y=8\\sin{t},\\text{slope}=-1\\hfill \\end{array}[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
\r\n
\r\n
\r\n
\r\n

12.\u00a0<\/strong>[latex]\\begin{array}{cc}x=t+\\frac{1}{t},\\hfill & y=t-\\frac{1}{t},\\text{slope}=1\\hfill \\end{array}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

\r\n

[reveal-answer q=\"540944\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"540944\"]No points possible; undefined expression.[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

\r\n
\r\n
13.\u00a0<\/strong>[latex]\\begin{array}{cc}x=2+\\sqrt{t},\\hfill & y=2 - 4t,\\text{slope}=0\\hfill \\end{array}[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n

For the following exercises, write the equation of the tangent line in Cartesian coordinates for the given parameter t<\/em>.<\/p>\r\n\r\n

\r\n
\r\n
\r\n
\r\n

14.\u00a0<\/strong>[latex]\\begin{array}{cc}x={e}^{\\sqrt{t}},\\hfill & y=1-\\text{ln}{t}^{2},t=1\\hfill \\end{array}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

\r\n

[reveal-answer q=\"64178\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"64178\"][latex]y=\\text{-}\\left(\\frac{4}{e}\\right)x+5[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

\r\n
\r\n
15.\u00a0<\/strong>[latex]\\begin{array}{cc}x=t\\text{ln}t,\\hfill & y={\\sin}^{2}t,\\hfill \\end{array}t=\\frac{\\pi }{4}[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
\r\n
\r\n
\r\n
\r\n

16.\u00a0<\/strong>[latex]\\begin{array}{cc}x={e}^{t},\\hfill & y={\\left(t - 1\\right)}^{2},\\text{at}\\left(1,1\\right)\\hfill \\end{array}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

\r\n

[reveal-answer q=\"922828\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"922828\"][latex]y=-2x + 3[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

\r\n
\r\n
17.\u00a0<\/strong>For [latex]x=\\sin\\left(2t\\right),y=2\\sin{t}[\/latex] where [latex]0\\le t<2\\pi [\/latex]. Find all values of t<\/em> at which a horizontal tangent line exists.<\/div>\r\n<\/div>\r\n<\/div>\r\n
\r\n
\r\n
\r\n
\r\n

18.\u00a0<\/strong>For [latex]x=\\sin\\left(2t\\right),y=2\\sin{t}[\/latex] where [latex]0\\le t<2\\pi [\/latex]. Find all values of t<\/em> at which a vertical tangent line exists.<\/p>\r\n\r\n<\/div>\r\n

\r\n

[reveal-answer q=\"25865\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"25865\"][latex]\\frac{\\pi }{4},\\frac{5\\pi }{4},\\frac{3\\pi }{4},\\frac{7\\pi }{4}[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

\r\n
\r\n
19.\u00a0<\/strong>Find all points on the curve [latex]x=4\\sin\\left(t\\right),y=4\\cos\\left(t\\right)[\/latex] that have the slope of [latex]0.5[\/latex].<\/div>\r\n<\/div>\r\n<\/div>\r\n
\r\n
\r\n
\r\n
\r\n

20.\u00a0<\/strong>Find [latex]\\frac{dy}{dx}[\/latex] for [latex]x=\\sin\\left(t\\right),y=\\cos\\left(t\\right)[\/latex].<\/p>\r\n\r\n<\/div>\r\n

\r\n

[reveal-answer q=\"106079\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"106079\"][latex]\\frac{dy}{dx}=\\text{-}\\tan\\left(t\\right)[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

\r\n
\r\n
21.\u00a0<\/strong>Find the equation of the tangent line to [latex]x=\\sin\\left(t\\right),y=\\cos\\left(t\\right)[\/latex] at [latex]t=\\frac{\\pi }{4}[\/latex].<\/div>\r\n<\/div>\r\n<\/div>\r\n
\r\n
\r\n
\r\n
\r\n

22.\u00a0<\/strong>For the curve [latex]x=4t,y=3t - 2[\/latex], find the slope and concavity of the curve at [latex]t=3[\/latex].<\/p>\r\n[reveal-answer q=\"580239\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"580239\"][latex]\\frac{dy}{dx}=\\frac{3}{4}[\/latex] and [latex]\\frac{{d}^{2}y}{d{x}^{2}}=0[\/latex], so the curve is neither concave up nor concave down at [latex]t=3[\/latex]. Therefore the graph is linear and has a constant slope but no concavity.[\/hidden-answer]\r\n\r\n<\/div>\r\n

<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n
\r\n
\r\n
23.\u00a0<\/strong>For the parametric curve whose equation is [latex]x=4\\cos\\theta ,y=4\\sin\\theta [\/latex], find the slope and concavity of the curve at [latex]\\theta =\\frac{\\pi }{4}[\/latex].<\/div>\r\n<\/div>\r\n<\/div>\r\n
\r\n
\r\n
\r\n
\r\n

24.\u00a0<\/strong>Find the slope and concavity for the curve whose equation is [latex]x=2+\\sec\\theta ,y=1+2\\tan\\theta [\/latex] at [latex]\\theta =\\frac{\\pi }{6}[\/latex].<\/p>\r\n\r\n<\/div>\r\n

\r\n

[reveal-answer q=\"265191\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"265191\"][latex]\\frac{dy}{dx}=4,\\frac{{d}^{2}y}{d{x}^{2}}=-6\\sqrt{3}[\/latex]; the curve is concave down at [latex]\\theta =\\frac{\\pi }{6}[\/latex].[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

\r\n
\r\n
25.\u00a0<\/strong>Find all points on the curve [latex]x=t+4,y={t}^{3}-3t[\/latex] at which there are vertical and horizontal tangents.<\/div>\r\n<\/div>\r\n<\/div>\r\n
\r\n
\r\n
\r\n
\r\n

26.\u00a0<\/strong>Find all points on the curve [latex]x=\\sec\\theta ,y=\\tan\\theta [\/latex] at which horizontal and vertical tangents exist.<\/p>\r\n\r\n<\/div>\r\n

\r\n

[reveal-answer q=\"247551\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"247551\"]No horizontal tangents. Vertical tangents at [latex]\\left(1,0\\right),\\left(-1,0\\right)[\/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 [latex]\\frac{{d}^{2}y}{d{x}^{2}}[\/latex].<\/p>\r\n\r\n

\r\n
\r\n
27.\u00a0<\/strong>[latex]\\begin{array}{cc}x={t}^{4}-1,\\hfill & y=t-{t}^{2}\\hfill \\end{array}[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
\r\n
\r\n
\r\n
\r\n

28.\u00a0<\/strong>[latex]\\begin{array}{cc}x=\\sin\\left(\\pi t\\right),\\hfill & y=\\cos\\left(\\pi t\\right)\\hfill \\end{array}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

\r\n

[reveal-answer q=\"196831\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"196831\"][latex]\\text{-}{\\sec}^{3}\\left(\\pi t\\right)[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

\r\n
\r\n
29. <\/strong>[latex]\\begin{array}{cc}x={e}^{\\text{-}t},\\hfill & y=t\\hfill \\end{array}{e}^{2t}[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n

For the following exercises, find points on the curve at which tangent line is horizontal or vertical.<\/p>\r\n\r\n

\r\n
\r\n
\r\n
\r\n

30.\u00a0<\/strong>[latex]\\begin{array}{cc}x=t\\left({t}^{2}-3\\right),\\hfill & y=3\\left({t}^{2}-3\\right)\\hfill \\end{array}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

\r\n

[reveal-answer q=\"660663\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"660663\"]Horizontal [latex]\\left(0,-9\\right)[\/latex]; vertical [latex]\\left(\\pm2,-6\\right)[\/latex].[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

\r\n
\r\n
31.\u00a0<\/strong>[latex]\\begin{array}{cc}x=\\frac{3t}{1+{t}^{3}},\\hfill & y=\\frac{3{t}^{2}}{1+{t}^{3}}\\hfill \\end{array}[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n

For the following exercises, find [latex]\\frac{dy}{dx}[\/latex] at the value of the parameter.<\/p>\r\n\r\n

\r\n
\r\n
\r\n
\r\n
\r\n

32.\u00a0<\/strong>[latex]\\begin{array}{cc}x=\\cos{t},\\hfill & y=\\sin{t},t=\\frac{3\\pi }{4}\\hfill \\end{array}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

\r\n

[reveal-answer q=\"692993\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"692993\"]1[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

33.\u00a0<\/strong>[latex]\\begin{array}{cc}x=\\sqrt{t},\\hfill & y=2t+4,t=9\\hfill \\end{array}[\/latex]<\/span><\/div>\r\n<\/div>\r\n<\/div>\r\n
\r\n
\r\n
\r\n
\r\n

34.\u00a0<\/strong>[latex]\\begin{array}{cc}x=4\\cos{(2\\pi s)},\\hfill & y=3\\sin{(2\\pi s)}\\hfill \\end{array} ,s=-\\frac{1}{4}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

\r\n

[reveal-answer q=\"393806\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"393806\"]0[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

For the following exercises, find [latex]\\frac{{d}^{2}y}{d{x}^{2}}[\/latex] at the given point without eliminating the parameter.<\/p>\r\n\r\n

\r\n
\r\n
35.\u00a0<\/strong>[latex]\\begin{array}{ccc}x=\\frac{1}{2}{t}^{2},\\hfill & y=\\frac{1}{3}{t}^{3},\\hfill & t=2\\hfill \\end{array}[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
\r\n
\r\n
\r\n
\r\n

36.\u00a0<\/strong>[latex]x=\\sqrt{t},y=2t+4,t=1[\/latex]<\/p>\r\n\r\n<\/div>\r\n

\r\n

[reveal-answer q=\"471769\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"471769\"]4[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

\r\n
\r\n
37.\u00a0<\/strong>Find t<\/em> intervals on which the curve [latex]x=3{t}^{2},y={t}^{3}-t[\/latex] is concave up as well as concave down.<\/div>\r\n<\/div>\r\n<\/div>\r\n
\r\n
\r\n
\r\n
\r\n

38.\u00a0<\/strong>Determine the concavity of the curve [latex]x=2t+\\text{ln}t,y=2t-\\text{ln}t[\/latex].<\/p>\r\n\r\n<\/div>\r\n

\r\n

[reveal-answer q=\"268180\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"268180\"]Concave up on [latex]t>0[\/latex].[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

\r\n
\r\n
39.\u00a0<\/strong>Sketch and find the area under one arch of the cycloid [latex]x=r\\left(\\theta -\\sin\\theta \\right),y=r\\left(1-\\cos\\theta \\right)[\/latex].<\/div>\r\n<\/div>\r\n<\/div>\r\n
\r\n
\r\n
\r\n
\r\n

40.\u00a0<\/strong>Find the area bounded by the curve [latex]x=\\cos{t},y={e}^{t},0\\le t\\le \\frac{\\pi }{2}[\/latex] and the lines [latex]y=1[\/latex] and [latex]x=0[\/latex].<\/p>\r\n\r\n<\/div>\r\n

\r\n

[reveal-answer q=\"28979\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"28979\"][latex]\\frac{e\\frac{1}{2}-1}{2}[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

\r\n
\r\n
41.\u00a0<\/strong>Find the area enclosed by the ellipse [latex]x=a\\cos\\theta ,y=b\\sin\\theta ,0\\le \\theta <2\\pi [\/latex].<\/div>\r\n<\/div>\r\n<\/div>\r\n
\r\n
\r\n
\r\n
\r\n

42.\u00a0<\/strong>Find the area of the region bounded by [latex]x=2{\\sin}^{2}\\theta ,y=2{\\sin}^{2}\\theta \\tan\\theta [\/latex], for [latex]0\\le \\theta \\le \\frac{\\pi }{2}[\/latex].<\/p>\r\n\r\n<\/div>\r\n

\r\n

[reveal-answer q=\"99568\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"99568\"][latex]\\frac{3\\pi }{2}[\/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 area of the regions bounded by the parametric curves and the indicated values of the parameter.<\/p>\r\n\r\n

\r\n
\r\n
43.\u00a0<\/strong>[latex]x=2\\cot\\theta ,y=2{\\sin}^{2}\\theta ,0\\le \\theta \\le \\pi [\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
\r\n
\r\n
\r\n
\r\n

44. [T]<\/strong> [latex]x=2a\\cos{t}-a\\cos\\left(2t\\right),y=2a\\sin{t}-a\\sin\\left(2t\\right),0\\le t<2\\pi [\/latex]<\/p>\r\n\r\n<\/div>\r\n

\r\n

[reveal-answer q=\"25958\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"25958\"][latex]6\\pi {a}^{2}[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

\r\n
\r\n
45. [T]<\/strong> [latex]x=a\\sin\\left(2t\\right),y=b\\sin\\left(t\\right),0\\le t<2\\pi [\/latex] (the \"hourglass\")<\/div>\r\n<\/div>\r\n<\/div>\r\n
\r\n
\r\n
\r\n
\r\n

46. [T]<\/strong> [latex]x=2a\\cos{t}-a\\sin\\left(2t\\right),y=b\\sin{t},0\\le t<2\\pi [\/latex] (the \"teardrop\")<\/p>\r\n\r\n<\/div>\r\n

\r\n

[reveal-answer q=\"960969\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"960969\"][latex]2\\pi ab[\/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 arc length of the curve on the indicated interval of the parameter.<\/p>\r\n\r\n

\r\n
\r\n
47.\u00a0<\/strong>[latex]x=4t+3,y=3t - 2,0\\le t\\le 2[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
\r\n
\r\n
\r\n
\r\n

48.\u00a0<\/strong>[latex]\\begin{array}{ccc}x=\\frac{1}{3}{t}^{3},\\hfill & y=\\frac{1}{2}{t}^{2},\\hfill & 0\\le t\\le 1\\hfill \\end{array}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

\r\n

[reveal-answer q=\"617959\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"617959\"][latex]\\frac{1}{3}\\left(2\\sqrt{2}-1\\right)[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

\r\n
\r\n
49.\u00a0<\/strong>[latex]\\begin{array}{ccc}x=\\cos\\left(2t\\right),\\hfill & y=\\sin\\left(2t\\right),\\hfill & 0\\le t\\le \\frac{\\pi }{2}\\hfill \\end{array}[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
\r\n
\r\n
\r\n
\r\n

50.\u00a0<\/strong>[latex]\\begin{array}{ccc}x=1+{t}^{2},\\hfill & y={\\left(1+t\\right)}^{3},\\hfill & 0\\le t\\le 1\\hfill \\end{array}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

\r\n

[reveal-answer q=\"867446\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"867446\"][latex]7.075[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

\r\n
\r\n
51.\u00a0<\/strong>[latex]\\begin{array}{ccc}x={e}^{t}\\cos{t},\\hfill & y={e}^{t}\\sin{t},\\hfill & 0\\le t\\le \\frac{\\pi }{2}\\hfill \\end{array}[\/latex] (express answer as a decimal rounded to three places)<\/div>\r\n<\/div>\r\n<\/div>\r\n
\r\n
\r\n
\r\n
\r\n

52.\u00a0<\/strong>[latex]x=a{\\cos}^{3}\\theta ,y=a{\\sin}^{3}\\theta [\/latex] on the interval [latex]\\left[0,2\\pi \\right)[\/latex] (the hypocycloid)<\/p>\r\n\r\n<\/div>\r\n

\r\n

[reveal-answer q=\"145058\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"145058\"][latex]6a[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

\r\n
\r\n
53.\u00a0<\/strong>Find the length of one arch of the cycloid [latex]x=4\\left(t-\\sin{t}\\right),y=4\\left(1-\\cos{t}\\right)[\/latex].<\/div>\r\n<\/div>\r\n<\/div>\r\n
\r\n
\r\n
\r\n
\r\n
\r\n

54.\u00a0<\/strong>Find the distance traveled by a particle with position [latex]\\left(x,y\\right)[\/latex] as t<\/em> varies in the given time interval: [latex]\\begin{array}{ccc}x={\\sin}^{2}t,\\hfill & y={\\cos}^{2}t,\\hfill & 0\\le t\\le 3\\pi \\hfill \\end{array}[\/latex].<\/p>\r\n\r\n<\/div>\r\n

\r\n

[reveal-answer q=\"418511\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"418511\"][latex]6\\sqrt{2}[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

55.\u00a0<\/strong>Find the length of one arch of the cycloid [latex]x=\\theta -\\sin\\theta ,y=1-\\cos\\theta [\/latex].<\/span><\/div>\r\n<\/div>\r\n<\/div>\r\n
\r\n
\r\n
56.\u00a0<\/strong>Show that the total length of the ellipse [latex]x=4\\sin\\theta ,y=3\\cos\\theta [\/latex] is [latex]L=16{\\displaystyle\\int }_{0}^{\\frac{\\pi}{2}}\\sqrt{1-{e}^{2}{\\sin}^{2}\\theta }d\\theta [\/latex], where [latex]e=\\frac{c}{a}[\/latex] and [latex]c=\\sqrt{{a}^{2}-{b}^{2}}[\/latex].<\/div>\r\n<\/div>\r\n<\/div>\r\n
\r\n
\r\n
57.\u00a0<\/strong>Find the length of the curve [latex]x={e}^{t}-t,y=4{e}^{\\frac{t}{2}},-8\\le t\\le 3[\/latex].<\/div>\r\n<\/div>\r\n<\/div>\r\n

For the following exercises, find the area of the surface obtained by rotating the given curve about the x<\/em>-axis.<\/p>\r\n\r\n

\r\n
\r\n
\r\n
\r\n

58.\u00a0<\/strong>[latex]\\begin{array}{ccc}x={t}^{3},\\hfill & y={t}^{2},\\hfill & 0\\le t\\le 1\\hfill \\end{array}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

\r\n

[reveal-answer q=\"574371\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"574371\"][latex]\\frac{2\\pi \\left(247\\sqrt{13}+64\\right)}{1215}[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

\r\n
\r\n
59.\u00a0<\/strong>[latex]\\begin{array}{ccc}x=a{\\cos}^{3}\\theta ,\\hfill & y=a{\\sin}^{3}\\theta ,\\hfill & 0\\le \\theta \\le \\hfill \\end{array}\\frac{\\pi }{2}[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
\r\n
\r\n
\r\n
\r\n

60. [T]<\/strong> Use a CAS to find the area of the surface generated by rotating [latex]x=t+{t}^{3},y=t-\\frac{1}{{t}^{2}},1\\le t\\le 2[\/latex] about the x<\/em>-axis. (Answer to three decimal places.)<\/p>\r\n\r\n<\/div>\r\n

\r\n

[reveal-answer q=\"724172\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"724172\"]59.101[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

\r\n
\r\n
61.\u00a0<\/strong>Find the surface area obtained by rotating [latex]x=3{t}^{2},y=2{t}^{3},0\\le t\\le 5[\/latex] about the y<\/em>-axis.<\/div>\r\n<\/div>\r\n<\/div>\r\n
\r\n
\r\n
\r\n
\r\n

62.\u00a0<\/strong>Find the area of the surface generated by revolving [latex]x={t}^{2},y=2t,0\\le t\\le 4[\/latex] about the x<\/em>-axis.<\/p>\r\n\r\n<\/div>\r\n

\r\n

[reveal-answer q=\"629866\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"629866\"][latex]\\frac{8\\pi }{3}\\left(17\\sqrt{17}-1\\right)[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

\r\n
\r\n
63. <\/strong>Find the surface area generated by revolving [latex]x={t}^{2},y=2{t}^{2},0\\le t\\le 1[\/latex] about the y<\/em>-axis.<\/div>\r\n<\/div>\r\n<\/div>","rendered":"

For the following exercises, each set of parametric equations represents a line. Without eliminating the parameter, find the slope of each line.<\/p>\n

\n
\n
1.\u00a0<\/strong>[latex]\\begin{array}{cc}x=3+t,\\hfill & y=1-t\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
\n

2. <\/strong>[latex]\\begin{array}{cc}x=8+2t,\\hfill & y=1\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n

\n

\n

Show Solution<\/span><\/p>\n
0<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
3.\u00a0<\/strong>[latex]\\begin{array}{cc}x=4 - 3t,\\hfill & y=-2+6t\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
\n

4.\u00a0<\/strong>[latex]\\begin{array}{cc}x=-5t+7,\\hfill & y=3t - 1\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n

\n

\n

Show Solution<\/span><\/p>\n
[latex]\\frac{-3}{5}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

For the following exercises, determine the slope of the tangent line, then find the equation of the tangent line at the given value of the parameter.<\/p>\n

\n
\n
\n
\n
\n

5.\u00a0<\/strong>[latex]\\begin{array}{cc}x=3\\sin{t},\\hfill & y=3\\cos{t},t=\\frac{\\pi }{4}\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

\n
\n
\n
\n

6.\u00a0<\/strong>[latex]\\begin{array}{cc}x=\\cos{t},\\hfill & y=8\\sin{t},\\hfill \\end{array}t=\\frac{\\pi }{2}[\/latex]<\/p>\n<\/div>\n

\n

\n

Show Solution<\/span><\/p>\n
[latex]\\text{Slope}=0[\/latex]; [latex]y=8[\/latex].<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
7.\u00a0<\/strong>[latex]\\begin{array}{cc}x=2t,\\hfill & y={t}^{3},t=-1\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
\n

8.\u00a0<\/strong>[latex]\\begin{array}{cc}x=t+\\frac{1}{t},\\hfill & y=t-\\frac{1}{t},t=1\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n

\n

\n

Show Solution<\/span><\/p>\n
Slope is undefined; [latex]x=2[\/latex].<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
\n
\n

9.\u00a0<\/strong>[latex]\\begin{array}{cc}x=\\sqrt{t},\\hfill & y=2t,t=4\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

For the following exercises, find all points on the curve that have the given slope.<\/p>\n

\n
\n
\n
\n

10.\u00a0<\/strong>[latex]\\begin{array}{cc}x=4\\cos{t},\\hfill & y=4\\sin{t},\\hfill \\end{array}[\/latex] slope = 0.5<\/p>\n<\/div>\n

\n

\n

Show Solution<\/span><\/p>\n
[latex]\\tan{t}=\\left(-2\\right)[\/latex] [latex]\\left(\\frac{4}{\\sqrt{5}},\\frac{-8}{\\sqrt{5}}\\right),\\left(\\frac{4}{\\sqrt{5}},\\frac{-8}{\\sqrt{5}}\\right)[\/latex].<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
11.\u00a0<\/strong>[latex]\\begin{array}{cc}x=2\\cos{t},\\hfill & y=8\\sin{t},\\text{slope}=-1\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
\n

12.\u00a0<\/strong>[latex]\\begin{array}{cc}x=t+\\frac{1}{t},\\hfill & y=t-\\frac{1}{t},\\text{slope}=1\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n

\n

\n

Show Solution<\/span><\/p>\n
No points possible; undefined expression.<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
13.\u00a0<\/strong>[latex]\\begin{array}{cc}x=2+\\sqrt{t},\\hfill & y=2 - 4t,\\text{slope}=0\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n<\/div>\n

For the following exercises, write the equation of the tangent line in Cartesian coordinates for the given parameter t<\/em>.<\/p>\n

\n
\n
\n
\n

14.\u00a0<\/strong>[latex]\\begin{array}{cc}x={e}^{\\sqrt{t}},\\hfill & y=1-\\text{ln}{t}^{2},t=1\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n

\n

\n

Show Solution<\/span><\/p>\n
[latex]y=\\text{-}\\left(\\frac{4}{e}\\right)x+5[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
15.\u00a0<\/strong>[latex]\\begin{array}{cc}x=t\\text{ln}t,\\hfill & y={\\sin}^{2}t,\\hfill \\end{array}t=\\frac{\\pi }{4}[\/latex]<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
\n

16.\u00a0<\/strong>[latex]\\begin{array}{cc}x={e}^{t},\\hfill & y={\\left(t - 1\\right)}^{2},\\text{at}\\left(1,1\\right)\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n

\n

\n

Show Solution<\/span><\/p>\n
[latex]y=-2x + 3[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
17.\u00a0<\/strong>For [latex]x=\\sin\\left(2t\\right),y=2\\sin{t}[\/latex] where [latex]0\\le t<2\\pi[\/latex]. Find all values of t<\/em> at which a horizontal tangent line exists.<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
\n

18.\u00a0<\/strong>For [latex]x=\\sin\\left(2t\\right),y=2\\sin{t}[\/latex] where [latex]0\\le t<2\\pi[\/latex]. Find all values of t<\/em> at which a vertical tangent line exists.<\/p>\n<\/div>\n

\n

\n

Show Solution<\/span><\/p>\n
[latex]\\frac{\\pi }{4},\\frac{5\\pi }{4},\\frac{3\\pi }{4},\\frac{7\\pi }{4}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
19.\u00a0<\/strong>Find all points on the curve [latex]x=4\\sin\\left(t\\right),y=4\\cos\\left(t\\right)[\/latex] that have the slope of [latex]0.5[\/latex].<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
\n

20.\u00a0<\/strong>Find [latex]\\frac{dy}{dx}[\/latex] for [latex]x=\\sin\\left(t\\right),y=\\cos\\left(t\\right)[\/latex].<\/p>\n<\/div>\n

\n

\n

Show Solution<\/span><\/p>\n
[latex]\\frac{dy}{dx}=\\text{-}\\tan\\left(t\\right)[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
21.\u00a0<\/strong>Find the equation of the tangent line to [latex]x=\\sin\\left(t\\right),y=\\cos\\left(t\\right)[\/latex] at [latex]t=\\frac{\\pi }{4}[\/latex].<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
\n

22.\u00a0<\/strong>For the curve [latex]x=4t,y=3t - 2[\/latex], find the slope and concavity of the curve at [latex]t=3[\/latex].<\/p>\n

Show Solution<\/span><\/p>\n
[latex]\\frac{dy}{dx}=\\frac{3}{4}[\/latex] and [latex]\\frac{{d}^{2}y}{d{x}^{2}}=0[\/latex], so the curve is neither concave up nor concave down at [latex]t=3[\/latex]. Therefore the graph is linear and has a constant slope but no concavity.<\/div>\n<\/div>\n<\/div>\n
<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
23.\u00a0<\/strong>For the parametric curve whose equation is [latex]x=4\\cos\\theta ,y=4\\sin\\theta[\/latex], find the slope and concavity of the curve at [latex]\\theta =\\frac{\\pi }{4}[\/latex].<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
\n

24.\u00a0<\/strong>Find the slope and concavity for the curve whose equation is [latex]x=2+\\sec\\theta ,y=1+2\\tan\\theta[\/latex] at [latex]\\theta =\\frac{\\pi }{6}[\/latex].<\/p>\n<\/div>\n

\n

\n

Show Solution<\/span><\/p>\n
[latex]\\frac{dy}{dx}=4,\\frac{{d}^{2}y}{d{x}^{2}}=-6\\sqrt{3}[\/latex]; the curve is concave down at [latex]\\theta =\\frac{\\pi }{6}[\/latex].<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
25.\u00a0<\/strong>Find all points on the curve [latex]x=t+4,y={t}^{3}-3t[\/latex] at which there are vertical and horizontal tangents.<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
\n

26.\u00a0<\/strong>Find all points on the curve [latex]x=\\sec\\theta ,y=\\tan\\theta[\/latex] at which horizontal and vertical tangents exist.<\/p>\n<\/div>\n

\n

\n

Show Solution<\/span><\/p>\n
No horizontal tangents. Vertical tangents at [latex]\\left(1,0\\right),\\left(-1,0\\right)[\/latex].<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

For the following exercises, find [latex]\\frac{{d}^{2}y}{d{x}^{2}}[\/latex].<\/p>\n

\n
\n
27.\u00a0<\/strong>[latex]\\begin{array}{cc}x={t}^{4}-1,\\hfill & y=t-{t}^{2}\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
\n

28.\u00a0<\/strong>[latex]\\begin{array}{cc}x=\\sin\\left(\\pi t\\right),\\hfill & y=\\cos\\left(\\pi t\\right)\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n

\n

\n

Show Solution<\/span><\/p>\n
[latex]\\text{-}{\\sec}^{3}\\left(\\pi t\\right)[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
29. <\/strong>[latex]\\begin{array}{cc}x={e}^{\\text{-}t},\\hfill & y=t\\hfill \\end{array}{e}^{2t}[\/latex]<\/div>\n<\/div>\n<\/div>\n

For the following exercises, find points on the curve at which tangent line is horizontal or vertical.<\/p>\n

\n
\n
\n
\n

30.\u00a0<\/strong>[latex]\\begin{array}{cc}x=t\\left({t}^{2}-3\\right),\\hfill & y=3\\left({t}^{2}-3\\right)\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n

\n

\n

Show Solution<\/span><\/p>\n
Horizontal [latex]\\left(0,-9\\right)[\/latex]; vertical [latex]\\left(\\pm2,-6\\right)[\/latex].<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
31.\u00a0<\/strong>[latex]\\begin{array}{cc}x=\\frac{3t}{1+{t}^{3}},\\hfill & y=\\frac{3{t}^{2}}{1+{t}^{3}}\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n<\/div>\n

For the following exercises, find [latex]\\frac{dy}{dx}[\/latex] at the value of the parameter.<\/p>\n

\n
\n
\n
\n
\n

32.\u00a0<\/strong>[latex]\\begin{array}{cc}x=\\cos{t},\\hfill & y=\\sin{t},t=\\frac{3\\pi }{4}\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n

\n

\n

Show Solution<\/span><\/p>\n
1<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
33.\u00a0<\/strong>[latex]\\begin{array}{cc}x=\\sqrt{t},\\hfill & y=2t+4,t=9\\hfill \\end{array}[\/latex]<\/span><\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
\n

34.\u00a0<\/strong>[latex]\\begin{array}{cc}x=4\\cos{(2\\pi s)},\\hfill & y=3\\sin{(2\\pi s)}\\hfill \\end{array} ,s=-\\frac{1}{4}[\/latex]<\/p>\n<\/div>\n

\n

\n

Show Solution<\/span><\/p>\n
0<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

For the following exercises, find [latex]\\frac{{d}^{2}y}{d{x}^{2}}[\/latex] at the given point without eliminating the parameter.<\/p>\n

\n
\n
35.\u00a0<\/strong>[latex]\\begin{array}{ccc}x=\\frac{1}{2}{t}^{2},\\hfill & y=\\frac{1}{3}{t}^{3},\\hfill & t=2\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
\n

36.\u00a0<\/strong>[latex]x=\\sqrt{t},y=2t+4,t=1[\/latex]<\/p>\n<\/div>\n

\n

\n

Show Solution<\/span><\/p>\n
4<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
37.\u00a0<\/strong>Find t<\/em> intervals on which the curve [latex]x=3{t}^{2},y={t}^{3}-t[\/latex] is concave up as well as concave down.<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
\n

38.\u00a0<\/strong>Determine the concavity of the curve [latex]x=2t+\\text{ln}t,y=2t-\\text{ln}t[\/latex].<\/p>\n<\/div>\n

\n

\n

Show Solution<\/span><\/p>\n
Concave up on [latex]t>0[\/latex].<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
39.\u00a0<\/strong>Sketch and find the area under one arch of the cycloid [latex]x=r\\left(\\theta -\\sin\\theta \\right),y=r\\left(1-\\cos\\theta \\right)[\/latex].<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
\n

40.\u00a0<\/strong>Find the area bounded by the curve [latex]x=\\cos{t},y={e}^{t},0\\le t\\le \\frac{\\pi }{2}[\/latex] and the lines [latex]y=1[\/latex] and [latex]x=0[\/latex].<\/p>\n<\/div>\n

\n

\n

Show Solution<\/span><\/p>\n
[latex]\\frac{e\\frac{1}{2}-1}{2}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
41.\u00a0<\/strong>Find the area enclosed by the ellipse [latex]x=a\\cos\\theta ,y=b\\sin\\theta ,0\\le \\theta <2\\pi[\/latex].<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
\n

42.\u00a0<\/strong>Find the area of the region bounded by [latex]x=2{\\sin}^{2}\\theta ,y=2{\\sin}^{2}\\theta \\tan\\theta[\/latex], for [latex]0\\le \\theta \\le \\frac{\\pi }{2}[\/latex].<\/p>\n<\/div>\n

\n

\n

Show Solution<\/span><\/p>\n
[latex]\\frac{3\\pi }{2}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

For the following exercises, find the area of the regions bounded by the parametric curves and the indicated values of the parameter.<\/p>\n

\n
\n
43.\u00a0<\/strong>[latex]x=2\\cot\\theta ,y=2{\\sin}^{2}\\theta ,0\\le \\theta \\le \\pi[\/latex]<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
\n

44. [T]<\/strong> [latex]x=2a\\cos{t}-a\\cos\\left(2t\\right),y=2a\\sin{t}-a\\sin\\left(2t\\right),0\\le t<2\\pi[\/latex]<\/p>\n<\/div>\n

\n

\n

Show Solution<\/span><\/p>\n
[latex]6\\pi {a}^{2}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
45. [T]<\/strong> [latex]x=a\\sin\\left(2t\\right),y=b\\sin\\left(t\\right),0\\le t<2\\pi[\/latex] (the “hourglass”)<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
\n

46. [T]<\/strong> [latex]x=2a\\cos{t}-a\\sin\\left(2t\\right),y=b\\sin{t},0\\le t<2\\pi[\/latex] (the “teardrop”)<\/p>\n<\/div>\n

\n

\n

Show Solution<\/span><\/p>\n
[latex]2\\pi ab[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

For the following exercises, find the arc length of the curve on the indicated interval of the parameter.<\/p>\n

\n
\n
47.\u00a0<\/strong>[latex]x=4t+3,y=3t - 2,0\\le t\\le 2[\/latex]<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
\n

48.\u00a0<\/strong>[latex]\\begin{array}{ccc}x=\\frac{1}{3}{t}^{3},\\hfill & y=\\frac{1}{2}{t}^{2},\\hfill & 0\\le t\\le 1\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n

\n

\n

Show Solution<\/span><\/p>\n
[latex]\\frac{1}{3}\\left(2\\sqrt{2}-1\\right)[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
49.\u00a0<\/strong>[latex]\\begin{array}{ccc}x=\\cos\\left(2t\\right),\\hfill & y=\\sin\\left(2t\\right),\\hfill & 0\\le t\\le \\frac{\\pi }{2}\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
\n

50.\u00a0<\/strong>[latex]\\begin{array}{ccc}x=1+{t}^{2},\\hfill & y={\\left(1+t\\right)}^{3},\\hfill & 0\\le t\\le 1\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n

\n

\n

Show Solution<\/span><\/p>\n
[latex]7.075[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
51.\u00a0<\/strong>[latex]\\begin{array}{ccc}x={e}^{t}\\cos{t},\\hfill & y={e}^{t}\\sin{t},\\hfill & 0\\le t\\le \\frac{\\pi }{2}\\hfill \\end{array}[\/latex] (express answer as a decimal rounded to three places)<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
\n

52.\u00a0<\/strong>[latex]x=a{\\cos}^{3}\\theta ,y=a{\\sin}^{3}\\theta[\/latex] on the interval [latex]\\left[0,2\\pi \\right)[\/latex] (the hypocycloid)<\/p>\n<\/div>\n

\n

\n

Show Solution<\/span><\/p>\n
[latex]6a[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
53.\u00a0<\/strong>Find the length of one arch of the cycloid [latex]x=4\\left(t-\\sin{t}\\right),y=4\\left(1-\\cos{t}\\right)[\/latex].<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
\n
\n

54.\u00a0<\/strong>Find the distance traveled by a particle with position [latex]\\left(x,y\\right)[\/latex] as t<\/em> varies in the given time interval: [latex]\\begin{array}{ccc}x={\\sin}^{2}t,\\hfill & y={\\cos}^{2}t,\\hfill & 0\\le t\\le 3\\pi \\hfill \\end{array}[\/latex].<\/p>\n<\/div>\n

\n

\n

Show Solution<\/span><\/p>\n
[latex]6\\sqrt{2}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
55.\u00a0<\/strong>Find the length of one arch of the cycloid [latex]x=\\theta -\\sin\\theta ,y=1-\\cos\\theta[\/latex].<\/span><\/div>\n<\/div>\n<\/div>\n
\n
\n
56.\u00a0<\/strong>Show that the total length of the ellipse [latex]x=4\\sin\\theta ,y=3\\cos\\theta[\/latex] is [latex]L=16{\\displaystyle\\int }_{0}^{\\frac{\\pi}{2}}\\sqrt{1-{e}^{2}{\\sin}^{2}\\theta }d\\theta[\/latex], where [latex]e=\\frac{c}{a}[\/latex] and [latex]c=\\sqrt{{a}^{2}-{b}^{2}}[\/latex].<\/div>\n<\/div>\n<\/div>\n
\n
\n
57.\u00a0<\/strong>Find the length of the curve [latex]x={e}^{t}-t,y=4{e}^{\\frac{t}{2}},-8\\le t\\le 3[\/latex].<\/div>\n<\/div>\n<\/div>\n

For the following exercises, find the area of the surface obtained by rotating the given curve about the x<\/em>-axis.<\/p>\n

\n
\n
\n
\n

58.\u00a0<\/strong>[latex]\\begin{array}{ccc}x={t}^{3},\\hfill & y={t}^{2},\\hfill & 0\\le t\\le 1\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n

\n

\n

Show Solution<\/span><\/p>\n
[latex]\\frac{2\\pi \\left(247\\sqrt{13}+64\\right)}{1215}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
59.\u00a0<\/strong>[latex]\\begin{array}{ccc}x=a{\\cos}^{3}\\theta ,\\hfill & y=a{\\sin}^{3}\\theta ,\\hfill & 0\\le \\theta \\le \\hfill \\end{array}\\frac{\\pi }{2}[\/latex]<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
\n

60. [T]<\/strong> Use a CAS to find the area of the surface generated by rotating [latex]x=t+{t}^{3},y=t-\\frac{1}{{t}^{2}},1\\le t\\le 2[\/latex] about the x<\/em>-axis. (Answer to three decimal places.)<\/p>\n<\/div>\n

\n

\n

Show Solution<\/span><\/p>\n
59.101<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
61.\u00a0<\/strong>Find the surface area obtained by rotating [latex]x=3{t}^{2},y=2{t}^{3},0\\le t\\le 5[\/latex] about the y<\/em>-axis.<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
\n

62.\u00a0<\/strong>Find the area of the surface generated by revolving [latex]x={t}^{2},y=2t,0\\le t\\le 4[\/latex] about the x<\/em>-axis.<\/p>\n<\/div>\n

\n

\n

Show Solution<\/span><\/p>\n
[latex]\\frac{8\\pi }{3}\\left(17\\sqrt{17}-1\\right)[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
63. <\/strong>Find the surface area generated by revolving [latex]x={t}^{2},y=2{t}^{2},0\\le t\\le 1[\/latex] about the y<\/em>-axis.<\/div>\n<\/div>\n<\/div>\n\n\t\t\t
\n\t\t\t

Candela Citations<\/h3>\n\t\t\t\t\t
\n\t\t\t\t\t\t
\n\t\t\t\t\t\t\t
CC licensed content, Shared previously<\/div>