Chapter 14.3, Problem 24E

### 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 4 ∫ 0 4 y − y 2 x 2 d x d y

To determine

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

Explanation

Given:

The integral:

âˆ«04âˆ«04yâˆ’y2x2dxdy

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‰¤yâ‰¤40â‰¤xâ‰¤4yâˆ’y2

The region bounded by these curves is shown below,

Since

0â‰¤xâ‰¤4yâˆ’y2,0â‰¤yâ‰¤4,

x=4yâˆ’y2x2=4yâˆ’y2r2cos2Î¸=4rsinÎ¸âˆ’r2sin2Î¸r2sin2Î¸+r2cos2Î¸=4rsinÎ¸

On simplifying further,

r2âˆ’4rsinÎ¸=0r(râˆ’4sinÎ¸)=0r=0,r=4sinÎ¸

As

y=0rsinÎ¸=0sin=0Î¸=0

And since,

y=4rsinÎ¸=44sinÎ¸Ã—sinÎ¸=4sin2Î¸=4sinÎ¸=Â±1

It is observed from the figure that the region lies in the first quadrant, therefore the corresponding limits in polar coordinates will be:

0â‰¤Î¸â‰¤Ï€20â‰¤râ‰¤4sinÎ¸

It should be noted that the limits can also be found from the figure above

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