Page 199 - How To Solve Word Problems In Calculus

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to x, from 0 to 1.

1

2
V = π y dx
0
1

2 2
= π (x − x ) dx
0
1

4
2
3
= π (x − 2x + x ) dx
0
3 4 5 1

x 2x x
= π − +
3 4 5 0

1 2 1
= π − + − 0
3 4 5
π
=
30

If the region bounded by two curves is revolved about the
x axis, the resulting solid of revolution will be hollow. Its cross
section will be in the shape of a washer, i.e., the area between
two concentric circles. The following method of computing
volume is named the washer method.
If the inner radius is y 1 and the outer radius is y 2 , and we
think of the thickness of the washer as dx, the volume of the
2
2
2
2
washer is V outer − V inner = πy dx − πy dx = π(y − y ) dx.
2
1
1
2
disk disk
y
2
y
1

The volume of the solid of revolution, expressed as an integral,

186