Page 203 - How To Solve Word Problems In Calculus

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EXAMPLE 8
Find the volume of the solid of revolution formed by rotating
2
the region bounded by the parabola y = x and the lines y = 0
and x = 2 about the x axis.

Solution

(2,4) (x, y)
4 4 h
dy
y h = 2 − x
r = y
(x,y) (2, y) 2

2 2

2
If (x, y) represents an arbitrary point on the graph y = x , the
length of the generated shell is h = 2 − x, the radius r = y, and
the thickness of the shell is dy. The volume of a typical shell
is 2πy(2 − x) dy and the total volume of the solid is obtained
by integration. Since the volume of the shell involves dy, in-
2
tegration will be with respect to y. Since y = x , y = 0 when
x = 0 and y = 4 when x = 2. Therefore,

4

V = 2π y (2 − x) dy
0

The variable x must be expressed in terms of y before the in-
2 √
tegration can be performed. Since y = x , x = y.

4

√
V = 2π y(2 − y) dy
0
4

= 2π (2y − y 3/2 ) dy
0

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