Chapter 15.3, Problem 40E

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8th Edition

James Stewart

ISBN: 9781285741550

Textbook Problem

(a) We define the improper integral (over the entire plane ℝ^{2})

where *D _{a}* is the disk with radius

(b) An equivalent definition of the improper integral in part (a) is

where *S _{a}* is the square with vertices (

(c) Deduce that

(d) By making the change of variable

(This is a fundamental result for probability and statistics.)

(a)

To determine

**To show:**

Explanation

**Given:**

The improper integral,

Here, *a.*

**Formula used:**

If *R* given by

If

**Calculation:**

From the given region *D,* it is observed that *r* varies from 0 to *a* and

Therefore, by the equation (1) the given integral becomes,

Integrate the function with respect to *r* and

Let

(b)

To determine

**To show:**

(c)

To determine

**To show:**

(d)

To determine

**To show:**

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