1.4E: Exercises for Section 1.4

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For exercises 1 - 3, find the length of the functions over the given interval.

1) $ y=5x$ from $ x=0$ to $ x=2$

Answer: $s = 2\sqrt{26}$ units

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

3) $ x=4y$ from $ y=−1$ to $ y=1$

Answer: $ s = 2\sqrt{17}$ units

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

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

$Coordinate axes with a translucent truncated cone lying along the X axis. The narrow end of the cone is at X equals 1, the wide end at X equals 4. The sides of the cone are slightly convex, and the upper edge has a solid line labeled 'Y equals square root of X' along it.$

Answer: $A = \frac{π}{6}(17\sqrt{17}−5\sqrt{5})$ units²

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

For exercises 7 - 16, find the lengths of the functions of $x$ over the given interval. If you cannot evaluate the integral exactly, use technology to approximate it.

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

Answer: $s= \frac{13\sqrt{13}−8}{27}$ units

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

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

Answer: $s= \frac{4}{3}$ units

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

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

Answer: $s \approx 2.0035$ units

12) $ y=\dfrac{x^3}{3}+\dfrac{1}{4x}$ from $ x=1$ to $ x=3$

13) $ y=\dfrac{x^4}{4}+\dfrac{1}{8x^2}$ from $ x=1$ to $ x=2$

Answer: $s= \frac{123}{32}$ units

14) $ y=\dfrac{2x^{3/2}}{3}−\dfrac{x^{1/2}}{2}$ from $ x=1$ to $ x=4$

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

Answer: $s=10$ units

16) [T] $ y=\sin x$ on $ x=0$ to $ x=π$

For exercises 17 - 26, find the lengths of the functions of $y$ over the given interval. If you cannot evaluate the integral exactly, use technology to approximate it.

17) $ y=\dfrac{5−3x}{4}$ from $ y=0$ to $ y=4$

Answer: $s= \frac{20}{3}$ units

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

19) $ x=5y^{3/2}$ from $ y=0$ to $ y=1$

Answer: $s= \frac{1}{675}(229\sqrt{229}−8)$ units

20) [T] $ x=y^2$ from $ y=0$ to $ y=1$

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

Answer: $s= \frac{1}{8}(4\sqrt{5}+\ln(9+4\sqrt{5}))$ units

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

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

Answer: $s \approx 1.201$ units

24) [T] $ x=\cos^2y$ from $ y=−\frac{π}{2}$ to $ y=\frac{π}{2}$

25) [T] $ x=4^y$ from $ y=0$ to $ y=2$

Answer: $s \approx 15.2341$ units

26) [T] $ x=\ln(y)$ on $ y=\dfrac{1}{e}$ to $ y=e$

For exercises 27 - 34, find the surface area of the volume generated when the following curves revolve around the $x$-axis. If you cannot evaluate the integral exactly, use your calculator to approximate it.

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

Answer: $A= \frac{49π}{3}$ units²

28) $ y=x^3$ from $ x=0$ to $ x=1$

29) $ y=7x$ from $ x=−1$ to $ x=1$

Answer: $A = 70π\sqrt{2}$ units²

30) [T] $ y=\frac{1}{x^2}$ from $ x=1$ to $ x=3$

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

Answer: $A = 8π$ units²

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

33) $ y=5x$ from $ x=1$ to $ x=5$

Answer: $A = 120π\sqrt{26}$ units²

34) [T] $ y=\tan x$ from $ x=−\frac{π}{4}$ to $ x=\frac{π}{4}$

For exercises 35 - 42, find the surface area of the volume generated when the following curves revolve around the $y$-axis. If you cannot evaluate the integral exactly, use your calculator to approximate it.

35) $ y=x^2$ from $ x=0$ to $ x=2$

Answer: $A= \frac{π}{6}(17\sqrt{17}−1)$ units²

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

37) $ y=x+1$ from $ x=0$ to $ x=3$

Answer: $A = 9\sqrt{2}π$ units²

38) [T] $ y=\dfrac{1}{x}$ from $ x=\dfrac{1}{2}$ to $ x=1$

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

Answer: $A = \frac{10\sqrt{10}π}{27}(73\sqrt{73}−1)$ units²

40) [T] $ y=3x^4$ from $ x=0$ to $ x=1$

41) [T] $ y=\dfrac{1}{\sqrt{x}}$ from $ x=1$ to $ x=3$

Answer: $A \approx 25.645$ units²

42) [T] $ y=\cos x$ from $ x=0$ to $ x=\frac{π}{2}$

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

$Translucent upper half of a sphere with a cylinder missing from its center. This shape sits in a coordinate system whose Y axis runs up the center of the missing cylinder. A solid line along the right profile of the hemisphere is labeled 'Y equals square root of 2 X minus X squared.'$

Answer: $A = 2π$ units²

44) A light bulb is a sphere with radius $1/2$ in. with the bottom sliced off to fit exactly onto a cylinder of radius $1/4$ in. and length $1/3$ 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 $1/4$ in. Find the surface area (not including the top or bottom of the cylinder).

$Side-by-side pictures of a translucent sphere above and nestled into the end of a cylinder, and beside it a screw-in light bulb.$

45) [T] A lampshade is constructed by rotating $ y=1/x$ around the $x$-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.

$Side-by-side pictures of a translucent truncated cone with slightly concave sides and narrow end at the top, and beside it a fabric lampshade with a similar shape.$

Answer: $10.5017$ units²

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

47) [T] You are building a bridge that will span $ 10$ ft. You intend to add decorative rope in the shape of $ y=5|\sin((xπ)/5)|$, where $ 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.

Answer: $ 23$ ft

48) Find the exact arc length of $ y=\ln(\sin x)$ from $ x=\frac{π}{4}$ to $ x=\frac{3π}{4}$. (Hint: Recall trigonometric identities.)

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

Answer: $2$

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

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

Answer: Answers may vary

52) [T] Which is longer between $(1,1)$ and $\left(2,\frac{1}{2}\right)$: the hyperbola $y=\dfrac{1}{x}$ or the graph of $x+2y=3$?

53) Explain why the surface area is infinite when $y=1/x$ is rotated around the $x$-axis for $ 1≤x<∞,$ but the volume is finite.

Answer: For more information, look up Gabriel’s Horn.

Contributors

Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at http://cnx.org.

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