   Chapter 15.3, Problem 24E

Chapter
Section
Textbook Problem

Use polar coordinates to find the volume of the given solid.24. Bounded by the paraboloid z = 1 + 2x2 + 2y2 and the plane z = 7 in the first octant

To determine

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

Explanation

Given:

The region D is bounded by the paraboloid z=1+2x2+2y2 and the plane z=7 in the first octant.

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:

Substitute z=7 in z=1+2x2+2y2 .

z=1+2x2+2y27=1+2x2+2y26=2x2+2y23=x2+y2

So, the value of r varies from 0 to 3 and the value of θ varies from 0 to π2 .

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

DzdA=0π203(7(1+2r2))(r)drdθ=0π203(6r2r3)drdθ

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

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