   Chapter 15.3, Problem 9E

Chapter
Section
Textbook Problem

Evaluate the given integral by changing to polar coordinates.9. ∬ R sin ( x 2 + y 2 )   d A , where R is the region in the first quadrant between the circles with center the origin and radii 1 and 3

To determine

To evaluate: The given integral by changing into the polar coordinates.

Explanation

Given:

The function, f(x,y)=sin(x2+y2) .

The region R lies between the circles centered at origin and the radii 1 and 3, respectively.

Formula used:

If f is a polar rectangle R given by 0arb,αθβ, where 0βα2π , then, Rf(x,y)dA=αβabf(rcosθ,rsinθ)rdrdθ (1)

If g(x) is the function of x and h(y) is the function of y then,

abcdg(x)h(y)dydx=abg(x)dxcdh(y)dy (2)

Calculation:

From the given region R, it is observed that r varies from 1 to 3 and θ varies from 0 to π2 .

Substitute x=rcosθ and y=rsinθ in the equation (1),

Rsin(x2+y2)dA=0π213sin(r2)rdrdθ

Integrate the function with respect to θ and r by using the equation (2).

0π213sin(r2)rdrdθ=13rsin(r2)dr0π2dθ

Let t=r2

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