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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

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BuyFindarrow_forward

Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
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)

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Chapter 15 Solutions

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