1.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{631b78c4-2763-4a1e-9d30-2f301fe3af2e-02_703_561_280_753}
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\caption{Figure 1}
\end{figure}
The shaded region R is bounded by the x -axis, the line with equation \(\mathrm { x } = 1\), the curve with equation \(y = 1 + \sqrt { x }\) and the y-axis, as shown in Figure 1. The unit of length on both of the axes is 1 m .
The region R is rotated through \(2 \pi\) radians about the x-axis to form a solid of revolution which is used to model a uniform solid \(S\).
Show, using the model and algebraic integration, that
- the volume of \(S\) is \(\frac { 17 \pi } { 6 } \mathrm {~m} ^ { 3 }\)
- the centre of mass of \(S\) is \(\frac { 49 } { 85 } \mathrm {~m}\) from 0 .
\includegraphics[max width=\textwidth, alt={}, center]{631b78c4-2763-4a1e-9d30-2f301fe3af2e-02_2264_41_314_1987}