Chapter 15.9, Problem 15E

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8th Edition

James Stewart

ISBN: 9781285741550

Textbook Problem

Use the given transformation to evaluate the integral.

**15.** ∫∫* _{R}* (

To determine

**To evaluate:** The integral

Explanation

**Given:**

The triangular region *R* with vertices

**Property used:** Change of Variable

Change of Variable in double integral is given by,

**Calculation:**

Obtain the Jacobian,

Find the partial derivative of *x* and *y* with respect to *u* and *v.*

From the given integral the function is, *x* and *y*.

Find the boundary by using the given transformation.

Find the line by using line equation

A corresponding line for the vertices

Substitute *x* and *y* in the given line,

A corresponding line for the vertices

Substitute *x* and *y* in the given line,

A corresponding line for the vertices

Substitute *x* and *y* in the given line,

In *uv* plane, the region is defines as follows,

By using equation (1) the integral will be,

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

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