   Chapter 14.3, Problem 63E

Chapter
Section
Textbook Problem

ProbabilityThe value of the integral I = ∫ − ∞ ∞ e − x 2 2 d x Is required in the normal probability density function.a) Use polar coordinates to evaluate the improper integral. I 2 = ( ∫ − ∞ ∞ e − x 2 2 d x ) ( ∫ − ∞ ∞ e − y 2 2 d y ) I 2 = ∫ − ∞ ∞ ∫ − ∞ ∞ e − ( x 2 + y 2 ) 2 d A b) Use the result of part (a) to determine I.

(a)

To determine

To calculate: The value of the integral I=ex22dx by converting to polar coordinates.

Explanation

Given:

I=ex22dx

Calculation:

First convert the integral into polar coordinates by substituting:

x=rcosθy=rsinθdxdy=rdrdθ

The corresponding limits in polar coordinates will be:

0θπ20r

Writing the double integral in polar coordinates:

I=ex22dxey22dy

I2=

(b)

To determine

To Calculate: The value of the integral by using results in a).

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