Chapter 14.3, Problem 18E

### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347

Chapter
Section

### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347
Textbook Problem

# Converting to Polar Coordinates:In Exercises 17–26, evaluate the iterated integral by converting to polar coordinates. ∫ 0 2 ∫ 0 4 − x 2 x d y d x

To determine

To Calculate: What the value of the iterated integral by converting to polar coordinates will be.

Explanation

Given:

The integral:

âˆ«02âˆ«04âˆ’x2xdydx

Formula Used:

The following formula of multiple integration is used in order to find the value of the iterated integral:

âˆ¬f(x)f(y)dxdy=âˆ«(âˆ«f(x)f(y)dx)dy

Calculation:

Convert the double integral into polar coordinates by substituting:

x=rcosÎ¸y=rsinÎ¸dxdy=rdrdÎ¸

The limits of the provided double integral are:

0â‰¤xâ‰¤20â‰¤yâ‰¤4âˆ’x2

The region bounded by these curves is shown below,

Since

x2=4âˆ’y2,y=2,

r2cos2Î¸=4âˆ’r2sin2Î¸,rsinÎ¸=2

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