22.38
Let SK. be the region bounded by y = x
3 + x, y = 0, and A: = 1 (Fig. 22-23). Find the volume of the solid
obtained by rotating &i about the y-axis.
Fig. 22-23
We use the cylindrical shell formula: V=2v JJ *(*
3 + x) dx = 2ir $„ (x
4 + x
2 ) dx = 2ir(^
5 + i*
3
) ]J =
2ir(\ + £)= 16-77/15.
22.42
Find the volume of the solid of rotation generated when the curve y = tan x, from x = 0 to x = IT 14, is
rotated about the jc-axis.
22.43 Find the volume of the solid generated by revolving about the x-axis the region bounded by y = sec x, y = 0,
x = 0, and x = ir/4.
See Fig. 22-24. By the disk formula, V= TT Jo"
4 sec
2 Jt dx = w(tan x) ]„'" = ir(l - 0) = IT.
The disk formula gives
22.37
Solve Problem 22.36 by the cylindrical shell formula.
VOLUME
181
We must integrate along the y-axis, and it is necessary to break the region into two pieces by the line y = 8.
Then we obtain
We have to find
Let
Then
Hence,
22.39
Same as Problem 22.38, but rotating about the x-axis.
We use the disk formula:
22.40
Same as Problem 22.38, but rotating about the line x = -t2.
22.41
Find the volume of the solid generated when one arch of the curve y = sin x, from x = 0 to x = IT, is
rotated about the jc-axis.
The disk formula yields
The same volume can be obtained by moving the region two units to the right and rotating about the _y-axis.
The new curve is y = (x - 2)
3 + x - 2, and the interval of integration is 2 < x< 3. By the cylindrical
shell formula,
V= 2n J 2
3 x[(x -2f + x-2}dx = 2Tr J 2
3 x(A:
3 - 6x
2 + 12x - 8 + x - 2) dx = 2ir J 2
3 (^
4 - 6:c
3 +
13A:
2 - 10^) dx = 277-G*
5 - Ix
4 + f x
3 - 5x
2 ) ] 2 = 2ir[( ^ - ^ + 117 - 45) - (f - 24 + ^ - 20)J = 617T/15.