2.
\begin{figure}[h]
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\caption{Figure 1}
\includegraphics[alt={},max width=\textwidth]{d366e541-15f6-4fb5-9afb-faf6120f1a1c-03_502_917_296_548}
\end{figure}
The curve with equation \(y = \frac { 1 } { 3 ( 1 + 2 x ) } , x > - \frac { 1 } { 2 }\), is shown in Figure 1.
The region bounded by the lines \(x = - \frac { 1 } { 4 } , x = \frac { 1 } { 2 }\), the \(x\)-axis and the curve is shown shaded in Figure 1.
This region is rotated through 360 degrees about the \(x\)-axis.
- Use calculus to find the exact value of the volume of the solid generated.
\begin{figure}[h]
\captionsetup{labelformat=empty}
\caption{Figure 2}
\includegraphics[alt={},max width=\textwidth]{d366e541-15f6-4fb5-9afb-faf6120f1a1c-03_383_447_1411_753}
\end{figure}
Figure 2 shows a paperweight with axis of symmetry \(A B\) where \(A B = 3 \mathrm {~cm}\). \(A\) is a point on the top surface of the paperweight, and \(B\) is a point on the base of the paperweight. The paperweight is geometrically similar to the solid in part (a). - Find the volume of this paperweight.