   Chapter 15.3, Problem 8E

Chapter
Section
Textbook Problem

Evaluate the given integral by changing to polar coordinates.8. ∬ R ( 2 x − y )   d A , where R is the region in the first quadrant enclosed by the circle x2 + y2 = 4 and the lines x = 0 and y = x

To determine

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

Explanation

Given:

The function, f(x,y)=2xy .

The region R is enclosed by the circle x2+y2=4 and the lines x=0,y=x .

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 0 to 2 and the value of θ varies from π4 to π2 .

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

R(2xy)dA=π4π202(2rcosθrsinθ)rdrdθ=π4π202r2(2cosθsinθ)drdθ

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

Show that the equation x4 + 4x + c = 0 has at most two real roots.

Single Variable Calculus: Early Transcendentals, Volume I

In Exercises 1520, simplify the expression. 15. 4(x2+y)3x2+y

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

In Problems 1-12, find the derivative of each function. 4.

Mathematical Applications for the Management, Life, and Social Sciences

For y = sin2 x + cos2 x, y = _____. a) 2 sin x 2 cos x b) 2 sin x cos x c) 4 sin x cos x d) 0

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th

For what values of p does the series converge?

Study Guide for Stewart's Multivariable Calculus, 8th 