Calculus Volume 1 · 6.4 Arc Length of a Curve and Surface Area

Arc Length of a Curve and Surface Area: practice problems

30 problems from the book, with its answers. Check your working, or have any of them explained to you on the whiteboard.

1. Exercises 165

   For the following exercises, find the length of the functions over the given interval.

   $y=5x\text{from}x=0\text{to}x=2$

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2. Exercises 167

   For the following exercises, find the length of the functions over the given interval.

   $x=4y\text{from}y=-1\text{to}y=1$

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3. Exercises 169

   For the following exercises, find the length of the functions over the given interval.

   Find the surface area of the volume generated when the curve $y=\sqrt{x}$ revolves around the $x\text{-axis}$ from $(1,1)$ to $(4,2),$ as seen here.

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4. Exercises 171

   For the following exercises, find the lengths of the functions of $x$ over the given interval. If you cannot evaluate the integral exactly, use technology to approximate it.

   $y=x^{3\text{/}2}$ from $(0,0)\text{to}(1,1)$

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5. Exercises 173

   For the following exercises, find the lengths of the functions of $x$ over the given interval. If you cannot evaluate the integral exactly, use technology to approximate it.

   $y=\frac{1}{3}{(x^2+2)}^{3\text{/}2}$ from

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6. Exercises 175

   For the following exercises, find the lengths of the functions of $x$ over the given interval. If you cannot evaluate the integral exactly, use technology to approximate it.

   [T] $y=e^x$ on $x=0$ to $x=1$

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7. Exercises 177

   For the following exercises, find the lengths of the functions of $x$ over the given interval. If you cannot evaluate the integral exactly, use technology to approximate it.

   $y=\frac{x^4}{4}+\frac{1}{8x^2}$ from

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8. Exercises 179

   For the following exercises, find the lengths of the functions of $x$ over the given interval. If you cannot evaluate the integral exactly, use technology to approximate it.

   $y=\frac{1}{27}{(9x^2+6)}^{3\text{/}2}$ from

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9. Exercises 181

   For the following exercises, find the lengths of the functions of $y$ over the given interval. If you cannot evaluate the integral exactly, use technology to approximate it.

   $y=\frac{5-3x}{4}$ from $y=0$ to $y=4$

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10. Exercises 183

    For the following exercises, find the lengths of the functions of $y$ over the given interval. If you cannot evaluate the integral exactly, use technology to approximate it.

    $x=5y^{3\text{/}2}$ from $y=0$ to $y=1$

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11. Exercises 185

    For the following exercises, find the lengths of the functions of $y$ over the given interval. If you cannot evaluate the integral exactly, use technology to approximate it.

    $x=\sqrt{y}$ from $y=0\text{to}y=1$

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12. Exercises 187

    For the following exercises, find the lengths of the functions of $y$ over the given interval. If you cannot evaluate the integral exactly, use technology to approximate it.

    [T] $x=\tan y$ from $y=0$ to $y=\frac{3}{4}$

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13. Exercises 189

    For the following exercises, find the lengths of the functions of $y$ over the given interval. If you cannot evaluate the integral exactly, use technology to approximate it.

    [T] $x=4^y$ from $y=0\text{to}y=2$

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14. Exercises 191

    For the following exercises, find the surface area of the volume generated when the following curves revolve around the $x\text{-axis}.$ If you cannot evaluate the integral exactly, use your calculator to approximate it.

    $y=\sqrt{x}$ from $x=2$ to $x=6$

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15. Exercises 193

    For the following exercises, find the surface area of the volume generated when the following curves revolve around the $x\text{-axis}.$ If you cannot evaluate the integral exactly, use your calculator to approximate it.

    $y=7x$ from $x=-1\text{to}x=1$

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16. Exercises 195

    For the following exercises, find the surface area of the volume generated when the following curves revolve around the $x\text{-axis}.$ If you cannot evaluate the integral exactly, use your calculator to approximate it.

    $y=\sqrt{4-x^2}$ from $x=0\text{to}x=2$

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17. Exercises 197

    For the following exercises, find the surface area of the volume generated when the following curves revolve around the $x\text{-axis}.$ If you cannot evaluate the integral exactly, use your calculator to approximate it.

    $y=5x$ from $x=1\text{to}x=5$

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18. Exercises 199

    For the following exercises, find the surface area of the volume generated when the following curves revolve around the $y\text{-axis}\text{.}$ If you cannot evaluate the integral exactly, use your calculator to approximate it.

    $y=x^2$ from $x=0\text{to}x=2$

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19. Exercises 201

    For the following exercises, find the surface area of the volume generated when the following curves revolve around the $y\text{-axis}\text{.}$ If you cannot evaluate the integral exactly, use your calculator to approximate it.

    $y=x+1$ from $x=0\text{to}x=3$

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20. Exercises 203

    For the following exercises, find the surface area of the volume generated when the following curves revolve around the $y\text{-axis}\text{.}$ If you cannot evaluate the integral exactly, use your calculator to approximate it.

    $y=\sqrt[3]{x}$ from $x=1\text{to}x=27$

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21. Exercises 205

    For the following exercises, find the surface area of the volume generated when the following curves revolve around the $y\text{-axis}\text{.}$ If you cannot evaluate the integral exactly, use your calculator to approximate it.

    [T] $y=\frac{1}{\sqrt{x}}$ from to

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22. Exercises 207

    For the following exercises, find the surface area of the volume generated when the following curves revolve around the $y\text{-axis}\text{.}$ If you cannot evaluate the integral exactly, use your calculator to approximate it.

    The base of a lamp is constructed by revolving a quarter circle $y=\sqrt{2x-x^2}$ around the from to as seen here. Create an integral for the surface area of this curve and compute it.

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23. Exercises 209

    For the following exercises, find the surface area of the volume generated when the following curves revolve around the $y\text{-axis}\text{.}$ If you cannot evaluate the integral exactly, use your calculator to approximate it.

    [T] A lampshade is constructed by rotating $y=1\text{/}x$ around the $x\text{-axis}$ from $y=1$ to $y=2,$ as seen here. Determine how much material you would need to construct this lampshade—that is, the surface area—accurate to four decimal places.

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24. Exercises 211

    For the following exercises, find the surface area of the volume generated when the following curves revolve around the $y\text{-axis}\text{.}$ If you cannot evaluate the integral exactly, use your calculator to approximate it.

    [T] You are building a bridge that will span $10$ ft. You intend to add decorative rope in the shape of $y=5|\sin ((x\pi )\text{/}5)|,$ where $x$ is the distance in feet from one end of the bridge. Find out how much rope you need to buy, measured in a whole number of feet.

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25. Exercises 213

    For the following exercises, find the exact arc length for the following problems over the given interval.

    [T] Draw graphs of $y=x^2,$ $y=x^6,$ and $y=x^{10}.$ For as increases, formulate a prediction on the arc length from to Now, compute the lengths of these three functions and determine whether your prediction is correct.

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26. Exercises 215

    For the following exercises, find the exact arc length for the following problems over the given interval.

    Solve for the length of $x=y^2$ from $(0,0)\text{to}(1,1).$ Show that $x=(1\text{/}2)y^2$ from $(0,0)$ to is twice as long. Graph both functions and explain why this is so.

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27. Exercises 217

    For the following exercises, find the exact arc length for the following problems over the given interval.

    Explain why the surface area is infinite when $y=1\text{/}x$ is rotated around the $x\text{-axis}$ for $1\le x<\infty ,$ but the volume is finite.

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28. Review Exercises 453

    For the following exercises, find the requested arc lengths.

    The length of $y$ for $x=3-\sqrt{y}$ from $y=0$ to

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29. Review Exercises 455

    For the following exercises, find the surface area and volume when the given curves are revolved around the specified axis.

    The loudspeaker created by revolving $y=1\text{/}x$ from $x=1$ to $x=4$ around the x-axis.

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30. Review Exercises 457

    For the following exercises, find the surface area and volume when the given curves are revolved around the specified axis.

    You are a crime scene investigator attempting to determine the time of death of a victim. It is noon and $45\text{°}\text{F}$ outside and the temperature of the body is $78\text{°}\text{F}.$ You know the cooling constant is $k=0.00824\text{°}\text{F/min}\text{.}$ When did the victim die, assuming that a human’s temperature is $98\text{°}\text{F}$

    ?

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