   Chapter 15.3, Problem 26E

Chapter
Section
Textbook Problem

Use polar coordinates to find the volume of the given solid.26. Bounded by the paraboloids z = 6 − x2 − y2 and z = 2x2 + 2y2

To determine

To find: The volume of the given solid by using polar coordinates.

Explanation

Given:

The region D is bounded by the paraboloids z=6x2y2 and z=2x2+2y2 .

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:

Equate both the paraboloids,

z=6x2y22x2+2y2=6x2y23x2+3y2=6x2+y2=2

So, the value of r varies from 0 to 2 and the value of θ varies from 0 to 2π .

Substitute x=rcosθ and y=rsinθ in the equation (1) and obtain the required volume

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