   Chapter 15, Problem 18RE

Chapter
Section
Textbook Problem

Describe the solid whose volume is given by the integral ∫ 0 π / 2 ∫ 0 π / 2 ∫ 1 2 ρ 2   sin   ϕ   d ρ   d ϕ   d θ and evaluate the integral.

To determine

To describe: The solid whose volume is given by the iterated integral and evaluate it.

Explanation

Given:

The region D is {(ρ,ϕ,θ)|1ρ2,0ϕπ2,0θπ2} .

Formula used:

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 (1)

Calculation:

Since ρ is varying between 1 and 2 and θ varies between 0 and π2 , the region lies between two spheres of radii 1 and 2 in the first octant. ϕ varies from 0 to π2 . So, the region lies above the curvature ϕ=π2 .

Use the equation (1) and separate the given iterated integral. Then, integrate it with respect to ϕ , θ and ρ

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