Page 201 - How To Solve Word Problems In Calculus
P. 201
P1: FYN/FZQ P2: FXS
PB056-07 Macros-FMT June 1, 2001 13:49 Char Count= 0
When the region is rotated, the outer radius of the washer, y 2 ,
is determined by the line and the inner radius, y 1 , by the
parabola.
b
2 2
V = π y − y 1 dx
2
a
2
2 2
2
= π [(2x) − (x ) ] dx
0
2
4
2
= π [4x − x ] dx
0
3 5 2
4x x
= π −
3 5
0
32 32
= π − − 0
3 5
64
= π
15
Another method for computing volumes of solids of revolu-
tion, the shell method, uses a different approach for construct-
ing an integral representing volume. Consider a thin-wall
cylindrical shell having inner and outer radii r 1 and r 2 , re-
spectively, where r 1 ≈ r 2 , and height h. (Imagine a soup can
with its top and bottom cut out. The thickness of the wall of
the can is r 2 − r 1 .)
h
r
2
r
1
The volume of the shell is the difference between two
cylindrical volumes.
188
