   Chapter 15.3, Problem 28E

Chapter
Section
Textbook Problem

(a) A cylindrical drill with radius r1 is used to bore a hole through the center of a sphere of radius r2. Find the volume of the ring-shaped solid that remains.(b) Express the volume in part (a) in terms of the height h of the ring. Notice that the volume depends only on h, not on r1 or r2.

(a)

To determine

To find: The volume of the given solid.

Explanation

Given:

The region D lies between the sphere of radius r22 and the cylinder of radius r12 .

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:

From the given region D, let z=r22x2y2 . And, r varies from r1 to r2 and θ varies from 0 to 2π .

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

DzdA=202πr1r2r22r2(r)drdθ=202πr1r2rr22r2drdθ

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

202πr1r2rr22r2drdθ=202πdθr1r2rr22r2dr

Let t=r2 .

Then, dt=2rdr .

Obtain the required volume as follows

(b)

To determine

To express: The volume obtained in part (a) in terms of height of the ring.

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