   Chapter 15.7, Problem 23E

Chapter
Section
Textbook Problem

Use cylindrical coordinates.23. Find the volume of the solid that is enclosed by the cone z = x 2 + y 2 and the sphere x2 + y2 + z2 = 2.

To determine

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

Explanation

Formula used:

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)

The cylindrical coordinates (r,θ,z) corresponding to the rectangular coordinates (x,y,z) is,

r=x2+y2θ=tan1(yx)z=z

Given:

The region E is enclosed by the cone z=x2+y2 and the sphere x2+y2+z2=2 .

Calculation:

From the given equations, the value of z is obtained below.

x2+y2+z2=2z2=2x2y2z=2x2y2z=2r2

And,

z=x2+y2z=r2z=r

It is observed that r varies from 0 to 1, and θ varies from 0 to 2π and z varies from r to z=2r2 .

Use the formula mentioned above to change the given problem into cylindrical coordinates. Then, by the equation (1), the volume of the given solid is computed as follows.

Ef(x,y,z)dV=0102πr2r2(r)dzdθdr=0102πr2r2rdzdθdr

Integrate with respect to z and apply the limit of it.

0102πr2r2rdzdθdr=0102πr[z]r2r2dθdr=0102πr(2r2r)dθdr=0102π(r2r2r2)dθdr=0102πr2r2dθdr0102πr2dθdr

Integrate with respect to θ and apply the limit of it

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