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2.5E: Exercises for Section 2.5

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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=226 units

2) y=12x+25 from x=1 to x=4

3) x=4y from y=1 to y=1

Answer
s=217 units

4) Pick an arbitrary linear function x=g(y) over any interval of your choice (y1,y2). 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=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=π6(171755) units2

6) Find the surface area of the volume generated when the curve y=x2 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=x3/2 from (0,0) to (1,1)

Answer
s=1313827 units

8) y=x2/3 from (1,1) to (8,4)

9) y=13(x2+2)3/2 from x=0 to x=1

Answer
s=43 units

10) y=13(x22)3/2 from x=2 to x=4

11) [T] y=ex on x=0 to x=1

Answer
s2.0035 units

12) y=x33+14x from x=1 to x=3

13) y=x44+18x2 from x=1 to x=2

Answer
s=12332 units

14) y=2x3/23x1/22 from x=1 to x=4

15) y=127(9x2+6)3/2 from x=0 to x=2

Answer
s=10 units

16) [T] y=sinx 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=53x4 from y=0 to y=4

Answer
s=203 units

18) x=12(ey+ey) from y=1 to y=1

19) x=5y3/2 from y=0 to y=1

Answer
s=1675(2292298) units

20) [T] x=y2 from y=0 to y=1

21) x=y from y=0 to y=1

Answer
s=18(45+ln(9+45)) units

22) x=23(y2+1)3/2 from y=1 to y=3

23) [T] x=tany from y=0 to y=34

Answer
s1.201 units

24) [T] x=cos2y from y=π2 to y=π2

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

Answer
s15.2341 units

26) [T] x=ln(y) on y=1e 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=x from x=2 to x=6

Answer
A=49π3 units2

28) y=x3 from x=0 to x=1

29) y=7x from x=1 to x=1

Answer
A=70π2 units2

30) [T] y=1x2 from x=1 to x=3

31) y=4x2 from x=0 to x=2

Answer
A=8π units2

32) y=4x2 from x=1 to x=1

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

Answer
A=120π26 units2

34) [T] y=tanx from x=π4 to x=π4

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=x2 from x=0 to x=2

Answer
A=π6(17171) units2

36) y=12x2+12 from x=0 to x=1

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

Answer
A=92π units2

38) [T] y=1x from x=12 to x=1

39) y=3x from x=1 to x=27

Answer
A=1010π27(73731) units2

40) [T] y=3x4 from x=0 to x=1

41) [T] y=1x from x=1 to x=3

Answer
A25.645 units2

42) [T] y=cosx from x=0 to x=π2

43) The base of a lamp is constructed by revolving a quarter circle y=2xx2 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=24ex/224, 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

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

48) y=ln(sinx) from x=π4 to x=3π4. (Hint: Recall trigonometric identities.)

49) Draw graphs of y=x2,y=x6, and y=x10. For y=xn, 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=y2 and the line x=by from (0,0) to (b2,b) as b increases. What do you notice?

51) Solve for the length of x=y2 from (0,0) to (1,1). Show that x=y22 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 (2,12): the hyperbola y=1x 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 1x<, 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.


This page titled 2.5E: Exercises for Section 2.5 is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to the style and standards of the LibreTexts platform.

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