   Chapter 15, Problem 40RE

Chapter
Section
Textbook Problem

Find the volume of the given solid.40. Above the paraboloid z = x2 + y2 and below the half-cone z =   x 2 +   y 2

To determine

To find: The volume of the given solid.

Explanation

Given:

The region E lies above the paraboloid z=x2+y2 and below the half cone z=x2+y2 .

Formula used:

The volume of the given solid is given by, V=EdV .

If f is a cylindrical region E given by h1(θ)rh2(θ),αθβ, u1(rcosθ,rsinθ)zu1(rcosθ,rsinθ) where 0βα2π , then,

Ef(x,y,z)dV=αβh1(θ)h2(θ)u1(rcosθ,rsinθ)u2(rcosθ,rsinθ)f(rcosθ,rsinθ,z)rdzdrdθ (1)

Calculation:

Since the equation of the cone is given, use of cylindrical coordinates is a wise choice.

Solve the given equations,

x2+y2=x2+y2x2+y2=x2+y2x2+y2x2+y2=1x2+y2=1

Therefore, r varies from 0 to 1, θ varies from 0 to 2π and z varies from x2+y2 to x2+y2 . In the cylindrical coordinate, x2+y2=r2 . Thus, z varies from r2 to r.

Then, by the equation (1), the volume of the given solid is,

V=EdV=02π01r2r(r)dzdrdθ

Integrate with respect to z and apply the limit

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