   Chapter 15, Problem 27RE

Chapter
Section
Textbook Problem

Calculate the value of the multiple integral.27. ∬ D ( x 2   +   y 2 ) 3 / 2   d A , where /9 is the region in the first quadrant bounded by the lines y = 0 and y   =   3   x and the circle x2 + y2 = 9

To determine

To calculate: The value of given double integral over the region R.

Explanation

Given:

The region D in the first quadrant bounded by the lines y=0 and y=3x and the circle x2+y2=9 .

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:

Since, the equation of the circle is given, it is more appropriate to use polar coordinates. From the given conditions, it is observed r varies from 0 to 3 and θ varies from 0 to π3 . Therefore, by the equation (1), the value of the double integral becomes,

D(x2+y2)32dA=0π303(x2+y2)3<

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