   Chapter 15.8, Problem 17E

Chapter
Section
Textbook Problem

Sketch the solid whose volume is given by the integral and evaluate the integral.17. ∫ 0 π / 6 ∫ 0 π / 2 ∫ 2 3 ρ2 sin ϕ dρ dθ dϕ

To determine

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

Explanation

Given:

The region D is {(ρ,ϕ,θ)|0ρ3,0ϕπ6,0θπ2} .

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 0 and 3 and θ varies between 0 and π2 , the region is the sphere of radius 3 in the first octant. ϕ varies from 0 to π6 . So, the outline sketch of the given region D is given below in the Figure 1.

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

0π60π203ρ2sinϕdρdθdϕ=0π6sinϕ</

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