1.4E: Exercises

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.

Answer

\(A = \frac{ \pi }{6}(17\sqrt{17}−5\sqrt{5})\) 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) [Technology Required] \( y=e^x\) 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) [Technology Required] \( y=\sin x\) on \( x=0\) to \( x= \pi \)

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) [Technology Required] \( x=y^2\) 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) [Technology Required] \( x=\tan y\) from \( y=0\) to \( y=\frac{3}{4}\)

Answer

\(s \approx 1.201\) units

24) [Technology Required] \( x=\cos^2y\) from \( y=−\frac{ \pi }{2}\) to \( y=\frac{ \pi }{2}\)

25) [Technology Required] \( x=4^y\) from \( y=0\) to \( y=2\)

Answer

\(s \approx 15.2341\) units

26) [Technology Required] \( x=\ln(y)\) 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

\(A= \frac{49 \pi }{3}\) units²

28) \( y=x^3\) from \( x=0\) to \( x=1\)

29) \( y=7x\) from \( x=−1\) to \( x=1\)

Answer

\(A = 70 \pi \sqrt{2}\) units²

30) [Technology Required] \( y=\frac{1}{x^2}\) from \( x=1\) to \( x=3\)

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

Answer

\(A = 8 \pi \) units²

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

33) \( y=5x\) from \( x=1\) to \( x=5\)

Answer

\(A = 120 \pi \sqrt{26}\) units²

34) [Technology Required] \( y=\tan x\) from \( x=−\frac{ \pi }{4}\) to \( x=\frac{ \pi }{4}\)

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

\(A= \frac{ \pi }{6}(17\sqrt{17}−1)\) 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

\(A = 9\sqrt{2} \pi \) units²

38) [Technology Required] \( y=\dfrac{1}{x}\) from \( x=\dfrac{1}{2}\) to \( x=1\)

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

Answer

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

40) [Technology Required] \( y=3x^4\) from \( x=0\) to \( x=1\)

41) [Technology Required] \( y=\dfrac{1}{\sqrt{x}}\) from \( x=1\) to \( x=3\)

Answer

\(A \approx 25.645\) units²

42) [Technology Required] \( y=\cos x\) from \( x=0\) to \( x=\frac{ \pi }{2}\)

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.

Answer

\(A = 2 \pi \) 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) [Technology Required] 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.

Answer

\(10.5017\) units²

46) [Technology Required] 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) [Technology Required] You are building a bridge that will span \( 10\) ft. You intend to add decorative rope in the shape of \( y=5|\sin((x \pi )/5)|\), 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

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

48) \( y=\ln(\sin x)\) from \( x=\frac{ \pi }{4}\) to \( x=\frac{3 \pi }{4}\). (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) [Technology Required] 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 \( 1 \leq x< \infty ,\) but the volume is finite.

Answer

For more information, look up Gabriel’s Horn.

In exercises 54 - 57, solve each problem.

54) [Technology Required] A chain hangs from two posts \(2\)m apart to form a catenary described by the equation \(y=2\cosh(x/2)−1\). Find the slope of the catenary at the left fence post.

Answer

\( -0.521095 \)

55) A telephone line is a catenary described by \(y=a\cosh(x/a).\) Find the ratio of the area under the catenary to its arc length. Does this confirm your answer for the previous question?

56) [Technology Required] A chain hangs from two posts four meters apart to form a catenary described by the equation \(y=4\cosh(x/4)−3.\) Find the total length of the catenary (arc length).

57) [Technology Required] A high-voltage power line is a catenary described by \(y=10\cosh(x/10)\). Find the ratio of the area under the catenary to its arc length. What do you notice?

Answer

\( 10 \)

58) [Technology Required] Find the arc length of \(y=1/x\) from \(x=1\) to \(x=4\).

59) [Technology Required] Find the surface area of the shape created when rotating the curve in the previous exercise from \(x=1\) to \(x=2\) around the \(x\)-axis.

60) [Technology Required] Find the arc length of \(\ln x\) from \(x=1\) to \(x=2\).

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.