# 6.4E: Exercises for Section 6.4

• • OpenStax
• OpenStax
$$\newcommand{\vecs}{\overset { \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$$$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

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

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

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

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

3) $$x=4y$$ from $$y=−1$$ to $$y=1$$

$$s = 2\sqrt{17}$$ 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.

$$A = \frac{π}{6}(17\sqrt{17}−5\sqrt{5})$$ 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 $$(1,1)$$

$$s= \frac{13\sqrt{13}−8}{27}$$ 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 $$x=1$$

$$s= \frac{4}{3}$$ 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 $$x=1$$

$$s \approx 2.0035$$ 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 $$x=2$$

$$s= \frac{123}{32}$$ 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 $$x=2$$

$$s=10$$ 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 $$y=4$$

$$s= \frac{20}{3}$$ 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 $$y=1$$

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

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

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

$$s= \frac{1}{8}(4\sqrt{5}+\ln(9+4\sqrt{5}))$$ 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 $$y=\frac{3}{4}$$

$$s \approx 1.201$$ 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 $$y=2$$

$$s \approx 15.2341$$ 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 $$x=6$$

$$A= \frac{49π}{3}$$ units2

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

29) $$y=7x$$ from $$x=−1$$ to $$x=1$$

$$A = 70π\sqrt{2}$$ units2

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

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

$$A = 8π$$ units2

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

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

$$A = 120π\sqrt{26}$$ 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 $$x=2$$

$$A= \frac{π}{6}(17\sqrt{17}−1)$$ 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 $$x=3$$

$$A = 9\sqrt{2}π$$ units2

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

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

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

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

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

$$A \approx 25.645$$ 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.

$$A = 2π$$ 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).

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.

$$10.5017$$ 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.

$$23$$ ft

For exercise 48, find the exact arc length for the following problems over the given interval.

48) $$y=\ln(\sin x)$$ 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.

$$2$$

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.

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.