Chapter 15, Problem 56RE

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8th Edition

James Stewart

ISBN: 9781285741550

Textbook Problem

Use the transformation *x* = *u*^{2}, *y* = *v*^{2}* z = w ^{2}* to find the volume of the region bounded by the surface

To determine

**To find:** The volume of the region bounded by the surface

Explanation

**Formula used:**

The Jacobian value is obtained by,

The volume of the region is

**Calculation:**

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

If

If

If

Compute the Jacobian value.

Rewrite the given transformations as,

Set

Therefore, *v* varies from 0 to
*w* varies from 0 to
*u* varies from 0 to 1.

Then, the given integral is,

Integrate the above integral with respect to *w* and apply the limit.

Simplify the above integral.

Integrate the above integral with respect to *v* and apply the limit

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