5.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f953fd1c-447f-4e97-a42a-5264c053fda0-10_684_689_260_639}
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\caption{Figure 1}
\end{figure}
Figure 1 shows part of the curve with equation \(y = e ^ { \frac { 1 } { 5 } x ^ { 2 } }\) for \(x \geq 0\)
The finite region \(R\), shown shaded in Figure 1, is bounded by the curve, the \(y\)-axis, the \(x\)-axis, and the line with equation \(x = 2\)
The table below shows corresponding values of \(x\) and \(y\) for \(y = e ^ { \frac { 1 } { 5 } x ^ { 2 } }\)
| \(x\) | 0 | 0.5 | 1 | 1.5 | 2 |
| \(y\) | 1 | \(e ^ { 0.05 }\) | \(e ^ { 0.2 }\) | \(e ^ { 0.45 }\) | \(e ^ { 0.8 }\) |
- Use the trapezium rule, with all the values of \(y\) in the table, to find an estimate for the area of \(R\), giving your answer to 2 decimal places.
- Use your answer to part (a) to deduce an estimate for
- \(\quad \int _ { 0 } ^ { 2 } \left( 4 + e ^ { \frac { 1 } { 5 } x ^ { 2 } } \right) d x\)
- \(\quad \int _ { 1 } ^ { 3 } e ^ { \frac { 1 } { 5 } ( x - 1 ) ^ { 2 } } d x\)
giving your answers to 2 decimal places.