   Chapter 14.3, Problem 27E

Chapter
Section
Textbook Problem

Converting to Polar Coordinates:In Exercises 27 and 28, combine the sum of the two iterated integrals into a single iterated integral by converting to polar coordinates. Evaluate the resulting iterated integral. ∫ 0 2 ∫ 0 x x 2 + y 2 d y d x + ∫ 2 2 2 ∫ 0 8 − x 2 x 2 + y 2 d y d x

To determine

To Calculate: The value of the sum of two iterated integrals when combined into a single

iterated integral by converting to polar coordinates.

Explanation

Given:

The integral:

020xx2+y2dydx+22208x2x2+y2dydx

Formula used:

The following formula is used to convert into polar coordinates:

Rf(x,y)dA=αβg1(θ)g2(θ)f(rcosθ,rsinθ)rdrdθ

Calculation:

Convert the double integral into polar coordinates by substituting:

x=rcosθy=rsinθdxdy=rdrdθ

The limits of the provided two double integrals are:

0x20yx

And

2x220y8x2

Since,

y=xrsinθ=rcosθtanθ=1θ=π4

As,

x=2rcosθ=2rcosπ4

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