Chapter 7.3, Problem 20E

To determine

To find: The volume of the solid.

Answer to Problem 20E

The volume of the given solid $V=8\pi$ .

Explanation of Solution

Given information:

The solid generated by revolving the region bounded by the lines and the curves:

  $\begin{array}{c}y=-\sqrt{x},\\ y=-2,\\ x=0\end{array}$

Calculation:

The limits of the integrations are $x=0$ and

  $x=4$ because those are the $x$ -coordinates of the intersection points of the curve $y=-\sqrt{x}$ and the lines $x=0$ and

  $y=-2$ respectively.

Each cross-section on height $x$ perpendicular to $x$ -axis is a circular region of radius $\text{2}$ with a circular region of radius $\sqrt{x}$ and the area of this cross-section is $A(x)=4\pi -x\pi$ .

The volume of the given solid is:

  

The graph is shown below:

  

Therefore, the required volume of the given solid $V=8\pi$ .