Chapter 15.3, Problem 10E

Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

Chapter
Section

Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
Textbook Problem

Evaluate the given integral by changing to polar coordinates.10. ∬ R y 2 x 2 + y 2 d A , where R is the region that lies between the circles x2 + y2 = a2 and x2 + y2 = b2 with 0 < a < b

To determine

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

Explanation

Given:

The function, f(x,y)=y2x2+y2 .

The region R lies between the circles centered at origin and the radii a and b, respectively with 0 < a < b.

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 the value of r varies from a to b and the value of θ varies from 0 to 2π Substitute x=rcosθ and y=rsinθ .

Therefore, by the equation (1) the given integral becomes,

Ry2x2+y2dA=02πabr2sin2θr2rdrdθ=02πabrsin2θdrdθ

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

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