Chapter 15.8, Problem 49E

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8th Edition

James Stewart

ISBN: 9781285741550

Textbook Problem

(a) Use cylindrical coordinates to show that the volume of the solid bounded above by the sphere *r ^{2}* +

(b) Deduce that the volume of the spherical wedge given by *ρ*_{1}
*ρ*
_{2},
_{,} *ϕ*_{1}
*ϕ*
*ϕ*_{2} is Δ*V* =
*(cosϕ*_{1 –} cos *ϕ*_{2})(

(c) Use the Mean Value Theorem to show that the volume in part (b) can be written as Δ*V* =
*ρ* Δ
*ϕ* where
*ρ*_{1} and *ρ*_{2},
*ϕ*_{1} and *ϕ*_{2}, Δ*ρ* = *ρ*_{2} – *ρ*_{1}, Δ=
_{2} –
_{2}, and Δ = *ϕ*_{2} – *ϕ*_{1}.

(a)

To determine

**To show:** The volume *V* of the given region is equal to

Explanation

**Formula used:**

The volume *V* of the given region *B* is,

If
*E* given by

If
*z*. Then,

The cylindrical coordinates

**Given:**

The region *B* bounded above by the sphere

**Calculation:**

It is given that
*z* varies from
*r*, Solve the given equations as shown below.

Thus, *r* varies from 0 to

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

Integrate with respect to *z* and apply the limit.

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

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