Chapter 15, Problem 13P

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8th Edition

James Stewart

ISBN: 9781285741550

Textbook Problem

The plane

To determine

**To find:** The volume of smaller piece when the plane cuts the ellipsoid.

Explanation

**Given:**

The equation of the plane is

The condition satisfied by the plane is

The equation of the ellipsoid is

**Formula used:**

The volume of the given region is

The Jacobian formula is,

**Calculation:**

Let the transformations be

Find the partial derivative of *x*, *y* and *z* with respect to *u*, *v* and *w*, respectively*.*

If

If

If

Obtain the Jacobian value.

Substitute the transformations in the ellipsoid of the region *E*,

The equation

To compute the volume of the sphere, consider the plane cuts the upper part of the sphere with the radius *r* and the smaller sliced part produces height *h*.

By using the spherical coordinates, the region of smaller slice of the sphere is obtained as,

Then, the volume of the integral is,

Integrate the above integral with respect to

Let

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