Chapter 14.3, Problem 29E

### 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 29–32, use polar coordinates to set up and evaluate the double integral ∫ R ∫ f ( x , y ) d A f ( x , y ) = x + y R : x 2 + y 2 ≤ 36 , x ≥ 0 , y ≥ 0

To determine

To Calculate: The equation of double integral and its value with the help of polar coordinates.

Explanation

Given:

The function:

f(x,y)=x+yR:x2+y2â‰¤36,xâ‰¥0,yâ‰¥0

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:

Using the provided data, first setup the double integral as:

âˆ«06âˆ«036âˆ’x2(x+y)dydx

Then, 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‰¤60â‰¤yâ‰¤36âˆ’x2

The region bounded by these curves is shown below,

Since

0â‰¤yâ‰¤36âˆ’x2,0â‰¤xâ‰¤6,

y=36âˆ’x2y2=36âˆ’x2r2sin2Î¸=36âˆ’r2cos2Î¸r2cos2Î¸+r2sin2Î¸=36r

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