   Chapter 14.3, Problem 24E

Chapter
Section
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:

0404yy2x2dxdy

Formula Used:

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

f(x)f(y)dxdy=(f(x)f(y)dx)dy

Calculation:

Convert the double integral into polar coordinates by substituting:

x=rcosθy=rsinθdxdy=rdrdθ

The limits of the provided double integral are:

0y40x4yy2

The region bounded by these curves is shown below,

Since

0x4yy2,0y4,

x=4yy2x2=4yy2r2cos2θ=4rsinθr2sin2θr2sin2θ+r2cos2θ=4rsinθ

On simplifying further,

r24rsinθ=0r(r4sinθ)=0r=0,r=4sinθ

As

y=0rsinθ=0sin=0θ=0

And since,

y=4rsinθ=44sinθ×sinθ=4sin2θ=4sinθ=±1

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

0θπ20r4sinθ

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

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