   Chapter 15.3, Problem 34E

Chapter
Section
Textbook Problem

Express the double integral in terms of a single integral with respect to r. Then use your calculator to evaluate the integral correct to four decimal places.34. ∬ D x y 1 + x 2 + y 2   d A , where D is the portion of the disk x2 + y2 ≤ 1 that lies in the first quadrant

To determine

To evaluate: The value of the double integral using calculator.

Explanation

Given:

The function is, z=xy1+x2+y2.

The region D is the portion of the disk x2+y21 that lies in the first quadrant.

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=xy1+x2+y2=(rcosθ)(rsinθ)1+r2=r2cosθsinθ1+r2

And, from the given region D, r varies from 0 to 1 and θ varies from 0 to π2.

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

DzdA=0π201r2cosθsinθ1+r2(r)drdθ=0π201r3cosθsinθ1+r2drdθ

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

Let t=sinθ

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