Page 197 - How To Solve Word Problems In Calculus

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b

= [ f (y) − g(y)] dy
a
2

2
= [(y + 2) − y ] dy
−1
y y
2 3 2

= + 2y −
2 3 −1

8 1 1
= 2 + 4 − − − 2 +
3 2 3

10 7
= − −
3 6
9
=
2

Volumes of Solids of Revolution
If the region bounded by the function y = f (x) and the x
axis, between x = a and x = b is revolved about the x axis, the
resulting three-dimensional figure is known as a solid of revo-
lution. Its cross-sectional area is circular, and its volume may
be computed by evaluating the integral

b

2
V = π [ f (x)] dx
a
b

2
or V = π y dx
a
A mnemonic device for remembering this formula is to think
of the solid being “sliced” into infinitesimally thin disks of
radius y and thickness dx. The volume of a typical disk is
2
2
π(radius) (thickness) = πy dx and the sum of the volumes is
b

2
π y dx. (Since π is a constant, it may be taken outside the
a
integral.) This method is sometimes known as the disk method.
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