9.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e30f0c28-1695-40a1-8e9a-6ea7e29042bf-16_727_1491_258_239}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\end{figure}
- By using the substitution \(u = 2 x + 3\), show that
$$\int _ { 0 } ^ { 12 } \frac { x } { ( 2 x + 3 ) ^ { 2 } } \mathrm {~d} x = \frac { 1 } { 2 } \ln 3 - \frac { 2 } { 9 }$$
The curve \(C\) has equation
$$y = \frac { 9 \sqrt { x } } { ( 2 x + 3 ) } , \quad x > 0$$
The finite region \(R\), shown shaded in Figure 3, is bounded by the curve \(C\), the \(x\)-axis and the line with equation \(x = 12\). The region \(R\) is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution.
- Use the result of part (a) to find the exact value of the volume of the solid generated.