   Chapter 14, Problem 26RE

Chapter
Section
Textbook Problem

Converting to Polar CoordinatesIn Exercises 25 and 26, evaluate the iterated integral by converting to polar coordinates. ∫ 0 4 ∫ 0 16 − y 2 ( x 2 + y 2 )   d x   d y

To determine

To calculate: The value of iterated integral 04016y2(x2+y2)dxdy by converting to polar coordinates

Explanation

Given:

The integral is 04016y2(x2+y2)dxdy

Formula used:

Area in polar coordinate is given as

dA=rdrdθ

Relation between rectangular and polar coordinate is given as

x2+y2=r2

Use the integration formula

xndx=xn+1n+1

Calculation:

x varies from 0 to 16y2. it represents a semicircle of radius 4 as

x=16y2

Squaring both the side:

x2=16y2

Add y2 on both side,

y2+x2=16y2+y2y2+x2=16

As r=x2+y2, therefore

r=4

y varies from 0 to 4 hence the resultant area is quadrant of the given circle

Plot the graph accordingly

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