   Chapter 5.R, Problem 11E

Chapter
Section
Textbook Problem

# Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. x 2 − y 2 = a 2 ,     x = a + h (where a>0, h>0); about the y -axis

To determine

To find:

The volume of a solid obtained by rotating the region bounded by the given curves about y-axis.

Explanation

1) Concept:

i. If x is the radius of a typical shell, then circumference=2πx and height is y

ii. By shell method, the volume of the solid by rotating the region under the curve y=f(x) about y- axis from a to b is

V= ab2πxf(x)dx

where,  0ab

2) Given:

The region bounded by x2-y2=a2,   x=a+h rotated about the y- axis (a>0, h>0).

The graph of x2-y2=a2 is a hyperbola with right and left branches.

Solving x2-y2=a2 for y,

y2=x2- a2

Taking square root on both the sides,

y= ±x2- a2

Using shell method, find the typical approximating shell with radius  x.

Therefore, the circumference is 2πx and height of each shell is

x2- a2    -   - x2- a2=2x2- a2

From the figure, the limits of integration is from  a to a+h.

So the total volume is

V= aa+h2πx[2x2- a2]    dx

Use substitution u=x2-a2

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