Page 217 - How To Solve Word Problems In Calculus

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The sphere is generated by rotating the upper half of the circle
2
2
x + y = 9 about the x axis. We solve this problem using the shell
method, using shells of radius y and length 2x.
b

V = 2π hr dr
a
3

= 2π 2xy dy
2
Since the variable of integration is y, we must replace x in terms of y.

2 2 2
Since x + y = 9, it follows that in the first quadrant x = 9 − y .
Hence
3

2
V = 2π 2y 9 − y dy
2
2
We evaluate this integral by making the substitution u = 9 − y .
It follows that du =−2ydy. We change the limits of integration
by observing that u = 5 when y = 2 and u = 0 when y = 3.
0

√
V =−2π udu
5
5

= 2π u 1/2 du
0
5
2
= 2π u 3/2
3
0
4 3/2
= π(5 − 0)
3
√
20π 5
=
3

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