Chapter 14.3, Problem 27E

### 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 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:

âˆ«02âˆ«0xx2+y2dydx+âˆ«222âˆ«08âˆ’x2x2+y2dydx

Formula used:

The following formula is used to convert into polar coordinates:

âˆ«Râˆ«f(x,y)dA=âˆ«Î±Î²âˆ«g1(Î¸)g2(Î¸)f(rcosÎ¸,rsinÎ¸)rdrdÎ¸

Calculation:

Convert the double integral into polar coordinates by substituting:

x=rcosÎ¸y=rsinÎ¸dxdy=rdrdÎ¸

The limits of the provided two double integrals are:

0â‰¤xâ‰¤20â‰¤yâ‰¤x

And

2â‰¤xâ‰¤220â‰¤yâ‰¤8âˆ’x2

Since,

y=xrsinÎ¸=rcosÎ¸tanÎ¸=1Î¸=Ï€4

As,

x=2rcosÎ¸=2rcosÏ€4

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