CHAPTER 31
Integrals for Surface Area,
Work, Centroids
SURFACE AREA OF A SOLID OF REVOLUTION
31.1
If the region under a curve y—f(x), above the x- axis, and between x = a and x = b, is revolved about
the *-axis, state a formula for the surface area S of the resulting solid.
31.4
The same arc as in Problem 31.3, but about the y-axis.
31.5
>> = A:
3
, Os* 253
[For revolution about the y-axis, change the factor y to x in either integrand.]
31.2
Find the surface area of a sphere of radius r.
Revolve the upper semicircle y =
about the jc-axis. Since
Hence, the surface area
S =
In Problems 31.3-31.13, find the surface area generated when the given arc is revolved about the given axis.
31.3
about the Jt-axis.
Hence, the surface area
Let x = i tan 0, dx = | sec
2 0 d6.
By the
reduction formula of Problem 29.39,
By Problem 29.40,
Thus we get
Since
So
so
31.6
about the *-axis.
31.7
in the first quadrant; about the x-axis.
So
In
In
In
we get
So
x~ + y
2 = r\ 2x+2yy' = Q,
y' = -x/y,
(y'Y = x*ly\
1 + (y') * = 1 + x'ly = (y- + x~)ly' = rly\
y = x
2 , 0 < x < j;
y' = 2x.
(sec
5 0-sec
3 e)de.
Use