   Chapter 15.3, Problem 31E

Chapter
Section
Textbook Problem

Evaluate the iterated integral by converting to polar coordinates.31. ∫ 0 1 / 2 ∫ 3 y 1 − y 2 x y 2   d x   d y

To determine

To evaluate: The iterated integral using polar coordinates.

Explanation

Given:

The function is, z=xy2 .

The variable x varies from 0 to 12 and y varies from 3y to 1y2 .

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:

In order to convert the given function into polar coordinates, substitute x=rcosθ and y=rsinθ . Thus, z becomes,

z=xy2=rcosθ(rsinθ)2=r3cosθsin2θ

Moreover, from the given condition of x and y, the value of r varies from 0 to 1 and the value of θ varies from 0 to π6 .

Therefore, by the equation (1), the value of the iterated integral becomes,

DzdA=0π601r3cosθsin2θ(r)drdθ=0π601r4cosθsin2θdrdθ

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

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