   Chapter 15.7, Problem 16E

Chapter
Section
Textbook Problem

Sketch the solid whose volume is given by the integral and evaluate the integral.16. ∫ 0 2 ∫ 0 2 π ∫ 0 r r   d z     d θ ​   d r

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)|0r2,0θ2π,0zr} .

Calculation:

Since r varies from 0 to 2 and θ varies from 0 to 2π , the region is the circle in the xy-plane with radius 2. z varies from 0 to r. 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.

0202π0rrdzdθdr=0202π[rz]0rdθdr=0202π[r(r)r(0)]dθdr=0202π[r20]dθdr=0202πr2dθdr

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

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