   Chapter 11, Problem 8PS

Chapter
Section
Textbook Problem

# Volume(a) Use the disk method to find the volume of the sphere x 2 + y 2 + z 2 = r 2 (b) Find the volume of the ellipsoid x 2 a 2 + y 2 b 2 + z 2 c 2 = 1 .

(a)

To determine

To calculate: Use the disk method and calculate the volume of the sphere x2+y2+z2=r2.

Explanation

Given:

The sphere’s equation is,

x2+y2+z2=r2

Formula used:

When the solid is revolved about the x-axis using the disk method the volume of a solid region will be,

V=x1x2π[R(x)]2dx

Calculation:

Consider the curve’s graph y=r2x2, x0 shown below.

The solid thus obtained is a hemi-sphere with the radius r when the curve is revolved about the x-axis.

Therefore, the volume of sphere using the disk method is,

V=2πx1x2[R(x)]2dx …… (1)

The limits for x is,

0xr

(b)

To determine

To calculate: The volume of the ellipsoid x2a2+y2b2+z2c2=1.

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