   Chapter 15, Problem 7RQ

Chapter
Section
Textbook Problem

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.If D is the disk given by x2 + y2 ≤ 4, then ∬ D 4   −   x 2   −   y 2   d A   =   16 3   π

To determine

Whether the statement, “If D is the disk given by x2+y24 then D4x2y2dA=163π ” is true or false.

Explanation

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)

Reason:

Convert the given problem into polar coordinates. For that, substitute x=rcosθ,y=rsinθ . Since, the given region is the disk of radius 2, r varies from 0 to 2 and θ varies from 0 to 2π . Then, by the equation (1), the value becomes,

D4x2y2dA=02π024r2(r)drdθ=02π02r4r2drdθ

Use the equation (2) to separate the integrals and integrate it. For that substitute t=r2,dt=2rdr

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