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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

Use polar coordinates to evaluate 0 3 9 x 2 9 x 2 ( x 3 + x y 2 ) d y d x

To determine

To evaluate: The value of the integral by using polar coordinates.

Explanation

Given:

The rectangular region is, R={(x,y)|0x3,9x2y9x2}.

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:

Substitute x=rcosθ,y=rsinθ in the given problem. From the given condition,

y=9x2y2=9x2x2+y2=9r2=9

From the above equation and from the given conditions, it is observed that r varies from 0 to 3 and θ varies from π2 to π2. Then, the value of the integral by (1) is,

039x29x2(x3+xy2)dydx=039x29

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Chapter 15 Solutions

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