   Chapter 15.8, Problem 49E

Chapter
Section
Textbook Problem

(a) Use cylindrical coordinates to show that the volume of the solid bounded above by the sphere r2 + z2 = a2 and below by the cone z = r cot ϕ0 (or ϕ = ϕ0), where 0 < ϕ0 < π/2, is V = 2 π a 3. 3 (1 – cosϕ0)(b) Deduce that the volume of the spherical wedge given by ρ1 ≤ ρ ≤ ρ2, θ 1 ≤ θ ≤ θ 2 , ϕ1 ≤ ϕ ≤ ϕ2 is ΔV = ρ 2 3 −   ρ 1 3 3 (cosϕ1 – cos ϕ2)( θ 2 – θ 1 )(c) Use the Mean Value Theorem to show that the volume in part (b) can be written as ΔV = ρ − 2 sin ϕ ¯ Δρ Δ θ Δϕ where ρ ¯ lies between ρ1 and ρ2, ϕ ¯ lies between ϕ1 and ϕ2, Δρ = ρ2 – ρ1, Δ= θ 2 – θ 2, and Δ = ϕ2 – ϕ1.

(a)

To determine

To show: The volume V of the given region is equal to 2πa33(1cosϕ0) by using cylindrical coordinates.

Explanation

Formula used:

The volume V of the given region B is, BdV .

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)

If g(x) is the function of x and h(y) is the function of y and k(z) is the function of z. Then, abcdefg(x)h(y)k(z)dzdydx=abg(x)dxcdh(y)dyefk(z)dz (2)

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 B bounded above by the sphere r2+z2=a2 and bounded below by the cone z=rcotϕ0 and ϕ0 varies from 0 to π2 .

Calculation:

It is given that θ varies from 0 to 2π ., and z varies from rcotϕ0 to a2r2 . To find the limits of r, Solve the given equations as shown below.

r2+z2=a2r2+r2cot2ϕ0=a2r2(1+cot2ϕ0)=a2r2a2=1csc2ϕ0

Thus, r varies from 0 to asinϕ0 .

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

BdV=02π0asinϕ0rcotϕ0a2r2dzdydx=02π0asinϕ0rcotϕ0a2r2rdzdrdθ

Integrate with respect to z and apply the limit.

02π0asinϕ0rcotϕ0a2r2rdzdrdq=02π0asinϕ0[rz]rcotϕ0a2r2drdθ=02π0asinϕ0[ra2r2(r)rcotϕ0]drdθ=02π0asinϕ0[ra2r2r2cotϕ0]drdθ=02π0asinϕ0ra2r2drdθ02π0asinϕ0r2cotϕ0drdθ

Use the equation (2) to separate the integral and integrate it

(b)

To determine

To deduce: The volume of the spherical wedge.

(c)

To determine

To show: The volume in part (b) can be written as ΔV=ρ¯2sinϕ¯ΔρΔθΔϕ by using mean value theorem.

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