   Chapter 14.3, Problem 29E

Chapter
Section
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+y236,x0,y0

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:

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

06036x2(x+y)dydx

Then, convert the double integral into polar coordinates by substituting:

x=rcosθy=rsinθdxdy=rdrdθ

The limits of the provided two double integrals are:

0x60y36x2

The region bounded by these curves is shown below,

Since

0y36x2,0x6,

y=36x2y2=36x2r2sin2θ=36r2cos2θr2cos2θ+r2sin2θ=36r

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