Area with parametric/substitution

9. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the curve with equation \(y = \mathrm { e } ^ { \sqrt { x } } , x > 0\) The finite region \(R\), shown shaded in Figure 1, is bounded by the curve, the \(x\)-axis and the lines \(x = 4\) and \(x = 9\)

1. Use the trapezium rule, with 5 strips of equal width, to obtain an estimate for the area of \(R\), giving your answer to 2 decimal places.
2. Use the substitution \(u = \sqrt { x }\) to find, by integrating, the exact value for the area of \(R\).

3. \begin{figure}[h] \end{figure} \section*{In this question you must show all stages of your working.} \section*{Solutions relying entirely on calculator technology are not acceptable.} Figure 1 shows a sketch of the curve with equation $$y = \sqrt { \frac { 3 x } { 3 x ^ { 2 } + 5 } } \quad x \geqslant 0$$ The finite region \(R\), shown shaded in Figure 1, is bounded by the curve, the \(x\)-axis and the lines with equations \(x = \sqrt { 5 }\) and \(x = 5\) The region \(R\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
Use integration to find the exact volume of the solid generated. Give your answer in the form \(a \ln b\), where \(a\) is an irrational number and \(b\) is a prime number.

5. \includegraphics[max width=\textwidth, alt={}, center]{5840974b-b08a-4818-9a59-97b2d3ce9890-1_469_809_1777_484} The diagram shows the curve with equation \(y = x \sqrt { 1 - x } , 0 \leq x \leq 1\).
Use the substitution \(u ^ { 2 } = 1 - x\) to show that the area of the region bounded by the curve and the \(x\)-axis is \(\frac { 4 } { 15 }\).