   Chapter 15.3, Problem 33E

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.33. ∬ D e ( x 2 + y 2 ) 2 d A , where D is the disk with center the origin and radius 1

To determine

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

Explanation

Given:

The function is, z=e(x2+y2)2.

The region D is the disk with center at origin and radius 1.

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=e(x2+y2)2=e(r2)2=er4

And, from the given region D, r varies from 0 to 1 and θ varies from 0 to 2π

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