[ill x = j .JY (y - 3),

aEXERCISES
1-4 Set up, but do not evaluate, an integral for the area of the
surface obtained by rotating the curve about (a) the x-axis and
(b) the y-axis.
rn y = x 4 , 0 ~ x ~ I 2. Y = xe-X, I ~ x ~ 3
4.x=~
5-12 Find the area of the surface obtained by rotating the curve
about the x-axis.
W y = x 3 , 0 ~ x ~ 2
6.9x=y2+18, 2~x~6
7. y = J"l""+4X, I ~ X ~ 5
8. y = cos 2x, 0 ~ X ~ 7T16
9. y = cosh x, 0 ~ x ~ I
x 3
I I
10. Y = - + -, :2 ~ x ~ I
6 2x
[IQX=5(y2 + 2)3/2, l~y~2
12.x=I+2y2, l~y~2
=21Tf: ~ du
13-16 The given curve is rotated about the y-axis. Find the area
of the resulting surface.
13. y = {j;, I ~ Y ~ 2
14. Y = I - x 2 , 0 ~ x ~ I
[ill x =~, 0 ~ Y ~ al2
16. y = ~X2 - ~ In x, I ~ x ~ 2
= 21T . Hsee () tan () + In 1see () + tan ()11:/4 (by Example 8 in Section 8.2)
= 1T[seea tan a + In(seea + tan a) -.fi -In(.fi + 1)]
s = 1T[e~ + In(e + ~) - .fi - In(.fi + 1)]
17-20 Use Simpson's Rule with n = 10 to approximate the area
of the surface obtained by rotating the curve about the x-axis.
Compare your answer with the value of the integral produced by
your calculator.
[ill] 21-22 Use either a CAS or a table of integrals to find the exact
area of the surface obtained by rotating the given curve about the
x-axis.
[ill] 23-24 Use a CAS to find the exact area of the surface obtained
by rotating the curve about the y-axis. If your CAS has trouble
evaluating the integral, express the surface area as an integral in
the other variable.
IThJ If the region rzJt = {(x, y) I x:;;' 1, 0 ~ Y ~ llx} is rotated
about the x-axis, the volume of the resulting solid is finite
(see Exercise 63 in Section 8.8). Show that the surface area is
infinite. (The surface is shown in the figure and is known as
Gabriel's horn.)
y