Chapter 8.3, Problem 25E

To determine

To find : the volume of the solid.

Answer to Problem 25E

The volume of the solid is $\frac{4\pi }{3}$ .

Explanation of Solution

Given information : The region enclosed about the y -axis and the triangle with vertices $\left(1,0\right),\left(2,1\right)\&\left(1,1\right)$ .

Calculation :

Area of each cross-sectional disk:

  $\begin{array}{l}A\left(y\right)=\pi \left(R^2-r^2\right)\\ \Rightarrow \pi \left[{\left(y+1\right)}^2-1^2\right]\\ \Rightarrow \pi \left(y^2+2y+1-1\right)\\ \Rightarrow \pi \left(y^2+2y\right)\end{array}$

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

  $\begin{array}{l}V={\displaystyle \int A\left(y\right)dy}\\ V={\displaystyle \underset{0}{\overset{1}{\int }}\pi \left(y^2+2y\right)dy}\\ V=\pi {\left[\frac{1}{3}{y}^3+y^2\right]}_0^1\\ V=\pi \left(\frac{1}{3}{\left(1\right)}^3+{\left(1\right)}^2\right)-\left(\frac{1}{3}{\left(0\right)}^3+{\left(0\right)}^2\right)\\ V=\pi \left[\frac{1}{3}+1\right]\\ V=\frac{4\pi }{3}\end{array}$