   Chapter 15.3, Problem 10E

Chapter
Section
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)

Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

Draw, in standard position, the angle whose measure is given. 21. 2 rad

Single Variable Calculus: Early Transcendentals, Volume I

In Exercises 1124, find the indicated limits, if they exist. 11. limx0 (5x 3)

Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach

If a X b, show that a E(X) b.

Probability and Statistics for Engineering and the Sciences 