   Chapter 14.3, Problem 23E

Chapter
Section
Textbook Problem

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

0202xx2xydydx

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:

0x20y2xx2

The region bounded by these curves is shown below,

Since

0y2xx2,0x2,

y=2xx2y2=2xx2r2sin2θ=2rcosθr2cos2θr2sin2θ+r2cos2θ=2rcosθ

On simplifying further,

r22rcosθ=0r(r2cosθ)=0r=0,r=2cosθ

As

x=0rcosθ=0cosθ

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