
# 1.3E: Exercises

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## Arc Length of a Curve and Surface Area

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

### Exercise $$\PageIndex{1}$$

$$\displaystyle y=5x$$ from $$\displaystyle x=0$$ to $$\displaystyle x=2$$

$$\displaystyle 2\sqrt{26}$$

### Exercise $$\PageIndex{2}$$

$$\displaystyle y=−\frac{1}{2}x+25$$ from $$\displaystyle x=1$$ to $$\displaystyle x=4$$

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### Exercise $$\PageIndex{3}$$

$$\displaystyle x=4y$$ from $$\displaystyle y=−1$$ to $$\displaystyle y=1$$

$$\displaystyle 2\sqrt{17}$$

### Exercise $$\PageIndex{4}$$

Pick an arbitrary linear function $$\displaystyle x=g(y)$$ over any interval of your choice $$\displaystyle (y_1,y_2).$$ Determine the length of the function and then prove the length is correct by using geometry.

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### Exercise $$\PageIndex{5}$$

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

$$\displaystyle \frac{π}{6}(17\sqrt{17}−5\sqrt{5})$$

### Exercise $$\PageIndex{6}$$

Find the surface area of the volume generated when the curve $$\displaystyle y=x^2$$ revolves around the $$\displaystyle y-axis$$ from $$\displaystyle (1,1)$$ to $$\displaystyle (3,9)$$.

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

### Exercise $$\PageIndex{7}$$

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

$$\displaystyle \frac{13\sqrt{13}−8}{27}$$

### Exercise $$\PageIndex{8}$$

$$\displaystyle y=x^{2/3}$$ from $$\displaystyle (1,1)$$ to $$\displaystyle (8,4)$$

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### Exercise $$\PageIndex{9}$$

$$\displaystyle y=\frac{1}{3}(x^2+2)^{3/2}$$ from $$\displaystyle x=0$$ to $$\displaystyle x=1$$

$$\displaystyle \frac{4}{3}$$

### Exercise $$\PageIndex{10}$$

$$\displaystyle y=\frac{1}{3}(x^2−2)^{3/2}$$ from $$\displaystyle x=2$$ to $$\displaystyle x=4$$

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### Exercise $$\PageIndex{11}$$

$$\displaystyle y=e^x$$ on $$\displaystyle x=0$$ to $$\displaystyle x=1$$

$$\displaystyle 2.0035$$

### Exercise $$\PageIndex{12}$$

$$\displaystyle y=\frac{x^3}{3}+\frac{1}{4x}$$ from $$\displaystyle x=1$$ to $$\displaystyle x=3$$

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### Exercise $$\PageIndex{13}$$

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

$$\displaystyle \frac{123}{32}$$

### Exercise $$\PageIndex{14}$$

$$\displaystyle y=\frac{2x^{3/2}}{3}−\frac{x^{1/2}}{2}$$ from $$\displaystyle x=1$$ to $$\displaystyle x=4$$

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### Exercise $$\PageIndex{15}$$

$$\displaystyle y=\frac{1}{27}(9x^2+6)^{3/2}$$ from $$\displaystyle x=0$$ to $$\displaystyle x=2$$

$$\displaystyle 10$$

### Exercise $$\PageIndex{16}$$

$$\displaystyle y=sinx$$ on $$\displaystyle x=0$$ to $$\displaystyle x=π$$

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

### Exercise $$\PageIndex{17}$$

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

$$\displaystyle \frac{20}{3}$$

### Exercise $$\PageIndex{18}$$

$$\displaystyle x=\frac{1}{2}(e^y+e^{−y})$$ from $$\displaystyle y=−1$$ to $$\displaystyle y=1$$

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### Exercise $$\PageIndex{19}$$

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

$$\displaystyle \frac{1}{675}(229\sqrt{229}−8)$$

### Exercise $$\PageIndex{20}$$

$$\displaystyle x=y^2$$ from $$\displaystyle y=0$$ to $$\displaystyle y=1$$

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### Exercise $$\PageIndex{21}$$

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

$$\displaystyle \frac{1}{8}(4\sqrt{5}+ln(9+4\sqrt{5}))$$

### Exercise $$\PageIndex{22}$$

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

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### Exercise $$\PageIndex{23}$$

$$\displaystyle x=tany$$ from $$\displaystyle y=0$$ to $$\displaystyle y=\frac{3}{4}$$

$$\displaystyle 1.201$$

### Exercise $$\PageIndex{24}$$

$$\displaystyle x=cos^2y$$ from $$\displaystyle y=−\frac{π}{2}$$ to $$\displaystyle y=\frac{π}{2}$$

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### Exercise $$\PageIndex{25}$$

$$\displaystyle x=4^y$$ from $$\displaystyle y=0$$ to $$\displaystyle y=2$$

$$\displaystyle 15.2341$$

### Exercise $$\PageIndex{26}$$

$$\displaystyle x=ln(y)$$ on $$\displaystyle y=\frac{1}{e}$$ to $$\displaystyle y=e$$

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

### Exercise $$\PageIndex{27}$$

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

$$\displaystyle \frac{49π}{3}$$

### Exercise $$\PageIndex{28}$$

$$\displaystyle y=x^3$$ from $$\displaystyle x=0$$ to $$\displaystyle x=1$$

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### Exercise $$\PageIndex{29}$$

$$\displaystyle y=7x$$ from $$\displaystyle x=−1$$ to $$\displaystyle x=1$$

$$\displaystyle 70π\sqrt{2}$$

### Exercise $$\PageIndex{30}$$

$$\displaystyle y=\frac{1}{x^2}$$ from $$\displaystyle x=1$$ to $$\displaystyle x=3$$

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### Exercise $$\PageIndex{31}$$

$$\displaystyle y=\sqrt{4−x^2}$$ from $$\displaystyle x=0$$ to $$\displaystyle x=2$$

$$\displaystyle 8π$$

### Exercise $$\PageIndex{32}$$

$$\displaystyle y=\sqrt{4−x^2}$$ from $$\displaystyle x=−1$$ to $$\displaystyle x=1$$

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### Exercise $$\PageIndex{33}$$

$$\displaystyle y=5x$$ from $$\displaystyle x=1$$ to $$\displaystyle x=5$$

$$\displaystyle 120π\sqrt{26}$$

### Exercise $$\PageIndex{34}$$

$$\displaystyle y=tanx$$ from $$\displaystyle x=−\frac{π}{4}$$ to $$\displaystyle x=\frac{π}{4}$$

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

### Exercise $$\PageIndex{35}$$

$$\displaystyle y=x^2$$ from $$\displaystyle x=0$$ to $$\displaystyle x=2$$

$$\displaystyle \frac{π}{6}(17\sqrt{17}−1)$$

### Exercise $$\PageIndex{36}$$

$$\displaystyle y=\frac{1}{2}x^2+\frac{1}{2}$$ from $$\displaystyle x=0$$ to $$\displaystyle x=1$$

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### Exercise $$\PageIndex{37}$$

$$\displaystyle y=x+1$$ from $$\displaystyle x=0$$ to $$\displaystyle x=3$$

$$\displaystyle 9\sqrt{2}π$$

### Exercise $$\PageIndex{38}$$

$$\displaystyle y=\frac{1}{x}$$ from $$\displaystyle x=\frac{1}{2}$$ to $$\displaystyle x=1$$

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### Exercise $$\PageIndex{39}$$

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

$$\displaystyle frac{10\sqrt{10}π}{27}(73\sqrt{73}−1)$$

### Exercise $$\PageIndex{40}$$

$$\displaystyle y=3x^4$$ from $$\displaystyle x=0$$ to $$\displaystyle x=1$$

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### Exercise $$\PageIndex{41}$$

$$\displaystyle y=\frac{1}{\sqrt{x}}$$ from $$\displaystyle x=1$$ to $$\displaystyle x=3$$

$$\displaystyle 25.645$$

### Exercise $$\PageIndex{42}$$

$$\displaystyle y=cosx$$ from $$\displaystyle x=0$$ to $$\displaystyle x=\frac{π}{2}$$

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### Exercise $$\PageIndex{43}$$

The base of a lamp is constructed by revolving a quarter circle $$\displaystyle y=\sqrt{2x−x^2}$$ around the $$\displaystyle y-axis$$ from $$\displaystyle x=1$$ to $$\displaystyle x=2$$, as seen here. Create an integral for the surface area of this curve and compute it.

$$\displaystyle 2π$$

### Exercise $$\PageIndex{44}$$

A light bulb is a sphere with radius $$\displaystyle 1/2$$ in. with the bottom sliced off to fit exactly onto a cylinder of radius $$\displaystyle 1/4$$ in. and length $$\displaystyle 1/3$$ in., as seen here. The sphere is cut off at the bottom to fit exactly onto the cylinder, so the radius of the cut is $$\displaystyle 1/4$$ in. Find the surface area (not including the top or bottom of the cylinder).

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### Exercise $$\PageIndex{45}$$

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

$$\displaystyle 10.5017$$

### Exercise $$\PageIndex{46}$$

An anchor drags behind a boat according to the function $$\displaystyle y=24e^{−x/2}−24$$, where $$\displaystyle y$$ represents the depth beneath the boat and $$\displaystyle x$$ is the horizontal distance of the anchor from the back of the boat. If the anchor is $$\displaystyle 23$$ ft below the boat, how much rope do you have to pull to reach the anchor? Round your answer to three decimal places.

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### Exercise $$\PageIndex{47}$$

You are building a bridge that will span $$\displaystyle 10$$ ft. You intend to add decorative rope in the shape of $$\displaystyle y=5|sin((xπ)/5)|$$, where $$\displaystyle x$$ is the distance in feet from one end of the bridge. Find out how much rope you need to buy, rounded to the nearest foot.

$$\displaystyle 23$$ ft

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

### Exercise $$\PageIndex{48}$$

$$\displaystyle y=ln(sinx)$$ from $$\displaystyle x=π/4$$ to $$\displaystyle x=(3π)/4$$. (Hint: Recall trigonometric identities.)

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### Exercise $$\PageIndex{49}$$

Draw graphs of $$\displaystyle y=x^2, y=x^6$$, and $$\displaystyle y=x^{10}$$. For $$\displaystyle y=x^n$$, as $$\displaystyle n$$ increases, formulate a prediction on the arc length from $$\displaystyle (0,0)$$ to $$\displaystyle (1,1)$$. Now, compute the lengths of these three functions and determine whether your prediction is correct.

$$\displaystyle 2$$

### Exercise $$\PageIndex{50}$$

Compare the lengths of the parabola $$\displaystyle x=y^2$$ and the line $$\displaystyle x=by$$ from $$\displaystyle (0,0)$$ to $$\displaystyle (b^2,b)$$ as $$\displaystyle b$$ increases. What do you notice?

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### Exercise $$\PageIndex{51}$$

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

### Exercise $$\PageIndex{52}$$

Which is longer between $$\displaystyle (1,1)$$ and $$\displaystyle (2,1/2)$$: the hyperbola $$\displaystyle y=1/x$$ or the graph of $$\displaystyle x+2y=3$$?

### Exercise $$\PageIndex{53}$$
Explain why the surface area is infinite when $$\displaystyle y=1/x$$ is rotated around the $$\displaystyle x-axis$$ for $$\displaystyle 1≤x<∞,$$ but the volume is finite.