   Chapter 15.7, Problem 15E

Chapter
Section
Textbook Problem

Sketch the solid whose volume is given by the integral and evaluate the integral.15. ∫ − π / 2 π / 2 ∫ 0 2 ∫ 0 r 2 r   d z   d r   d θ

To determine

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

Explanation

Given:

The region D is {(r,θ,z)|0r2,π2θπ2,0zr2} .

Calculation:

Since r varies between 0 and 2 and θ varies between π2 and π2 , the region is the semicircle in the xy-plane with radius 2. z varies from 0 to r2 . So, the outline sketch of the given region D is given below in the Figure 1.

Integrate the given iterated integral with respect to z and apply the limit of it.

π2π2020r2rdzdrdθ=π2π202[rz]0r2drdθ=π2π202[r(r2)r(0)]drdθ=π2π202[r30]drdθ=π2π202r3drdθ

Integrate the given iterated integral with respect to r and apply the limit of it

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