Area under curve requiring parts

5. \begin{figure}[h] \end{figure} Figure 1 shows the graph of the curve with equation $$y = x \mathrm { e } ^ { 2 x } , \quad x \geqslant 0$$ The finite region \(R\) bounded by the lines \(x = 1\), the \(x\)-axis and the curve is shown shaded in Figure 1.

1. Use integration to find the exact value for the area of \(R\).
2. Complete the table with the values of \(y\) corresponding to \(x = 0.4\) and 0.8 .

   \(x\)	0	0.2	0.4	0.6	0.8	1
   \(y = x \mathrm { e } ^ { 2 x }\)	0	0.29836		1.99207		7.38906

3. Use the trapezium rule with all the values in the table to find an approximate value for this area, giving your answer to 4 significant figures.

5. \begin{figure}[h] \end{figure}

1. Find \(\int \frac { \ln x } { x ^ { 2 } } \mathrm {~d} x\) Figure 3 shows a sketch of part of the curve with equation $$y = \frac { 3 + 2 x + \ln x } { x ^ { 2 } } \quad x > 0.5$$ The finite region \(R\), shown shaded in Figure 3, is bounded by the curve, the line with equation \(x = 2\), the \(x\)-axis and the line with equation \(x = 4\)
2. Use the answer to part (a) to find the exact area of \(R\), writing your answer in simplest form.

12. \begin{figure}[h] \end{figure} The figure 6 shows a sketch of the curve with equation $$y = x ^ { 2 } \ln 2 x$$ The finite region \(R\), shown shaded in figure 5, is bounded by the line with equation \(x = \frac { e ^ { 2 } } { 2 }\), the curve \(C\), the line with equation \(x = e ^ { 2 }\) and the \(x\)-axis.
Show that the exact value of the area of region \(R\) is \(\frac { e ^ { 6 } } { 72 } ( 35 + 24 \ln 2 )\).

11. \begin{figure}[h] \end{figure} In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. Figure 5 shows a sketch of part of the curve \(C\) with equation $$y = 8 x ^ { 2 } \mathrm { e } ^ { - 3 x } \quad x \geqslant 0$$ The finite region \(R\), shown shaded in Figure 5, is bounded by

• the curve \(C\)
• the line with equation \(x = 1\)
• the \(x\)-axis

Find the exact area of \(R\), giving your answer in the form $$A + B \mathrm { e } ^ { - 3 }$$ where \(A\) and \(B\) are rational numbers to be found.

14 The region bounded by the curve $$y = ( 2 x - 8 ) \ln x$$ and the \(x\)-axis is shaded in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{22ff390e-1360-43bd-8c7f-3d2b58627e91-26_867_908_543_566} 14

1. Use the trapezium rule with 5 ordinates to find an estimate for the area of the shaded region. Give your answer correct to three significant figures.
   14
2. Show that the exact area is given by $$32 \ln 2 - \frac { 33 } { 2 }$$ Fully justify your answer.