# What is the volume obtained by rotating the region enclosed by y=11-x, y=3x+7, and x=0 about the y-axis?

Mar 27, 2016

$V = {V}_{1} - {V}_{2} = 11 , 525 \frac{\pi}{27}$

#### Explanation:

Given: $y = 11 - x$, $y = 3 x + 7$, and $x = 0$
Required: The volume obtained by rotating about the y-axis
Solution strategy: looking at the figure we see that

$V = 2 \pi \left[{\int}_{1}^{11} x \left(11 - x\right) \mathrm{dx} - {\int}_{- 2 \frac{1}{3}}^{1} x \left(3 x + 7\right) \mathrm{dx}\right]$

${V}_{1} = 2 \pi \left[{\left(\frac{11}{2} {x}^{2} - {x}^{3} / 3\right)}_{1}^{11}\right]$
${V}_{2} = 2 \pi \left[{\left({x}^{3} + 7 {x}^{2} / 2\right)}_{- 2 \frac{1}{3}}^{1}\right]$
$V = {V}_{1} - {V}_{2} = 11 , 525 \frac{\pi}{27}$