Chapter 7.3, Problem 7E

To determine

To find: The volume of the solid generated by revolving the shaded region about the given axis.

Answer to Problem 7E

The answer: $V=\frac{2\pi }{3}$

Explanation of Solution

Given information:

The figure is:

About $x$

-the axis:

  

Calculation:

The region in the picture is bounded by the lines:

  $\begin{array}{c}x=0,\\ y=0\\ y=-\frac{1}{2}x+1\end{array}$

Also, from the picture conclude that the limits of integration are:

  $\begin{array}{c}x=0\\ x=2\end{array}$

Consider the solid produced by rotating the specified region about the $x$ -axis now.

each cross-section on height $x$ perpendicular to the $x$ - axis is a circle with radius $-\frac{1}{2}x+1$ and area

  $A(x)={\left(-\frac{1}{2}x+1\right)}^2\pi$ .

So that, the volume of the solid is given by:

  

  $\begin{array}{c}={\left.\pi \left(\frac{x^3}{12}-\frac{x^2}{2}+x\right)\right|}_0^2\\ =\frac{2\pi }{3}\end{array}$

Therefore, the required volume of the given solid is equal to $V=\frac{2\pi }{3}$ .