12 The shaded region shown in the diagram below is bounded by the \(x\)-axis, the curve \(y = \mathrm { f } ( x )\), and the lines \(x = a\) and \(x = b\)
\includegraphics[max width=\textwidth, alt={}, center]{74b8239a-1f46-45e7-ad20-2dce7bf4baf6-16_661_721_406_662}
The shaded region is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid.
12
- Show that the volume of this solid is
$$\pi \int _ { a } ^ { b } ( \mathrm { f } ( x ) ) ^ { 2 } \mathrm {~d} x$$
12
- In the case where \(a = 1 , b = 2\) and
$$f ( x ) = \frac { x + 3 } { ( x + 1 ) \sqrt { x } }$$
show that the volume of the solid is
$$\pi \left( \ln \left( \frac { 2 ^ { m } } { 3 ^ { n } } \right) - \frac { 2 } { 3 } \right)$$
where \(m\) and \(n\) are integers.