Complete table then estimate

4. $$y = \frac { 5 } { \left( x ^ { 2 } + 1 \right) }$$

1. Complete the table below, giving the missing value of \(y\) to 3 decimal places.

   \(x\)	0	0.5	1	1.5	2	2.5	3
   \(y\)	5	4	2.5		1	0.690	0.5

   (1) \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{1c51b071-5cb1-4841-b031-80bde9027433-06_732_1118_826_411} \captionsetup{labelformat=empty} \caption{Figure 1}
   \end{figure} Figure 1 shows the region \(R\) which is bounded by the curve with equation \(y = \frac { 5 } { \left( x ^ { 2 } + 1 \right) }\),
   the \(x\)-axis and the lines \(x = 0\) and \(x = 3\) the \(x\)-axis and the lines \(x = 0\) and \(x = 3\)
2. Use the trapezium rule, with all the values of \(y\) from your table, to find an approximate value for the area of \(R\).
3. Use your answer to part (b) to find an approximate value for $$\int _ { 0 } ^ { 3 } \left( 4 + \frac { 5 } { \left( x ^ { 2 } + 1 \right) } \right) d x$$ giving your answer to 2 decimal places.

3. (a) \(\quad y = 5 ^ { x } + \log _ { 2 } ( x + 1 ) , \quad 0 \leqslant x \leqslant 2\) Complete the table below, by giving the value of \(y\) when \(x = 1\) (b) Use the trapezium rule, with all the values of \(y\) from the completed table, to find an approximate value for $$\int _ { 0 } ^ { 2 } \left( 5 ^ { x } + \log _ { 2 } ( x + 1 ) \right) \mathrm { d } x$$ giving your answer to 2 decimal places.
(c) Use your answer to part (b) to find an approximate value for $$\int _ { 0 } ^ { 2 } \left( 5 + 5 ^ { x } + \log _ { 2 } ( x + 1 ) \right) d x$$ giving your answer to 2 decimal places.

1. $$y = \sqrt { \left( 2 ^ { x } + x \right) }$$ a. Complete the table below, giving the values of \(y\) to 3 decimal places. (1)
b. Use the trapezium rule with all the values of \(y\) from your table to find an approximation for the value of $$\int _ { 0 } ^ { 1 } \sqrt { \left( 2 ^ { x } + x \right) } \mathrm { d } x$$ giving your answer to 3 significant figures. Using your answer to part (b) and making your method clear, estimate
c. \(\int _ { 0 } ^ { 0.5 } \sqrt { \left( 2 ^ { 2 x } + 2 x \right) } \mathrm { d } x\)

\includegraphics{figure_2} Figure 2 shows the curve with equation $$y = x^2 \sin\left(\frac{1}{2}x\right), \quad 0 < x \leq 2\pi.$$ The finite region \(R\) bounded by the line \(x = \pi\), the \(x\)-axis, and the curve is shown shaded in Fig 2.

1. Find the exact value of the area of \(R\), by integration. Give your answer in terms of \(\pi\). [7]

The table shows corresponding values of \(x\) and \(y\).

1. Find the value of \(G\). [1]
2. Use the trapezium rule with values of \(x^2 \sin\left(\frac{1}{2}x\right)\)
   1. at \(x = \pi\), \(x = \frac{3\pi}{2}\) and \(x = 2\pi\) to find an approximate value for the area \(R\), giving your answer to 4 significant figures,
   2. at \(x = \pi\), \(x = \frac{5\pi}{4}\), \(x = \frac{3\pi}{2}\), \(x = \frac{7\pi}{4}\) and \(x = 2\pi\) to find an improved approximation for the area \(R\), giving your answer to 4 significant figures.
   [5]