Chapter 8.3, Problem 27E

To determine

To find : the volume of the solid.

Answer to Problem 27E

The volume of the solid is $8\pi$ .

Explanation of Solution

Given information : The region in the first quadrant bounded above by the parabola $y=x^2$ , below by the x -axis, and on the right by the line $x=2$ .

Calculation :

Volume of the solid obtained after rotating the given region about y -axis is:

  $\begin{array}{l}V={\displaystyle \int \pi \left(R^2-r^2\right)}\\ V={\displaystyle \underset{0}{\overset{4}{\int }}\pi {\left(2\right)}^2-{\sqrt{y}}^{2}dy}\\ V=\pi {\displaystyle \underset{0}{\overset{4}{\int }}4-ydy}\\ V=\pi {\left(4y-\frac{y^2}{2}\right)}_0^4\\ V=\pi \left[4\left(4\right)-\frac{{\left(4\right)}^2}{2}-0\right]\\ V=8\pi \end{array}$