Answer
a) See image
b) $\int_{0}^{3} \pi(x^4+10x^2+25) dx $
c) $\frac{1068\pi}{5}$
Work Step by Step
a) img
b)
$\\volume = \int\pi(y)^2$ apply $\ y=x^2+5$ to formula,
$\ \pi(x^2+5)^2 = \pi(x^4+10x^2+25)$ shaded region is 0 to 3, so $\int_{0}^{3} \pi(x^4+10x^2+25) dx $
c)
1) Apply Sum Rule of Integration\(\displaystyle \pi \int _{0}^{3}x^{4}+10x^{2}+25~dx\)
\(\displaystyle \pi (\int _{0}^{3}x^{4}~dx+\int _{0}^{3}10x^{2}~dx+\int _{0}^{3}25~dx)\)
2)Apply the Constant Multiple Rule of Integration
\(\displaystyle \pi (\int _{0}^{3}x^{4}~dx+\int _{0}^{3}{10}x^{2}~dx+\int _{0}^{3}{25}~dx)\)
\(\displaystyle \pi (\int _{0}^{3}x^{4}~dx+{10}\int _{0}^{3}x^{2}~dx+{25}\int _{0}^{3}1~dx)\)
3)Apply the Power Rule of Integration\(\displaystyle \pi (\int _{0}^{3}x^{4}~dx+10\int _{0}^{3}x^{2}~dx+25\int _{0}^{3}1~dx)\)\(\displaystyle \pi \left[\frac{x^{4+1}}{4+1}+10\cdot \frac{x^{2+1}}{2+1}+25\cdot \frac{x^{0+1}}{0+1}\right]_{0}^{3}\)
4)Simplify
\(\displaystyle \pi \left[\frac{1}{5}x^{5}+\frac{10}{3}x^{3}+25x\right]_{0}^{3}\)
5)Substitute and subtract\(\displaystyle \pi \left[\frac{1}{5}x^{5}+\frac{10}{3}x^{3}+25x\right]_{0}^{3}\)
\(\pi \left(\frac{1}{5}\cdot 3^{5}+\frac{10}{3}\cdot 3^{3}+25\cdot 3-\left(\frac{1}{5}\cdot 0^{5}+\frac{10}{3}\cdot 0^{3}+25\cdot 0\right)\right)\)
6)Simplify
\(\pi \cdot \frac{1068}{5}\)
