Answer
$163.02712$
Work Step by Step
a) Given
$$x^2-y^2=7, x=4 ; \quad \text { about } y=5$$
First , we find the intersection points
\begin{aligned}
x^2-y^2&=7\\
16-y^2&=7\\
y^2-9&=0\\
y&=\pm 3
\end{aligned}
Since the volume of the generated solid given by
\begin{aligned}
V&=2\pi \int_a^b r(y)h(y)dy
\end{aligned}
Here
$$ h(y) =4- \sqrt{7+y^2},\ \ r(y)= 5-y$$
Then
\begin{aligned}
V&=2\pi \int_a^b r(y)h(y)dy\\
&= 2\pi \int_{3-}^{3} \left(5-y\right)(4- \sqrt{7+y^2})dy
\end{aligned}
b) Using the calculator, we get
\begin{aligned}
V &=2\pi \int_{-3}^{3} \left(5-y\right)(4- \sqrt{7+y^2})dy \\
&\approx 163.02712
\end{aligned}
