Complete table then estimate

4. (a) Sketch the graph of \(y = \frac { 1 } { x } , x > 0\) The table below shows corresponding values of \(x\) and \(y\) for \(y = \frac { 1 } { x }\), with the values for \(y\) rounded to 3 decimal places where necessary. (b) Use the trapezium rule with all the values of \(y\) from the table to find an approximate value, to 2 decimal places, for \(\int _ { 1 } ^ { 3 } \frac { 1 } { x } \mathrm {~d} x\)

4. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the curve with equation \(y = \sqrt { x + 2 } , x \geqslant - 2\) The finite region \(R\), shown shaded in Figure 1, is bounded by the curve, the \(x\)-axis and the line \(x = 6\) The table below shows corresponding values of \(x\) and \(y\) for \(y = \sqrt { x + 2 }\)

1. Complete the table above, giving the missing value of \(y\) to 4 decimal places.
2. Use the trapezium rule, with all of the values of \(y\) in the completed table, to find an approximate value for the area of \(R\), giving your answer to 3 decimal places. Use your answer to part (b) to find approximate values of
3. 1. \(\int _ { - 2 } ^ { 6 } \frac { \sqrt { x + 2 } } { 2 } \mathrm {~d} x\)
   2. \(\int _ { - 2 } ^ { 6 } ( 2 + \sqrt { x + 2 } ) \mathrm { d } x\)

1. The table below shows corresponding values of \(x\) and \(y\) for \(y = \frac { 1 } { \sqrt { ( x + 1 ) } }\), with the values
   for \(y\) rounded to 3 decimal places where necessary.

1. Complete the table by giving the value of \(y\) corresponding to \(x = 15\)
2. Use the trapezium rule with all the values of \(y\) from the completed table to find an approximate value for $$\int _ { 0 } ^ { 15 } \frac { 1 } { \sqrt { ( x + 1 ) } } \mathrm { d } x$$ giving your answer to 2 decimal places.

2. $$y = \frac { 2 ^ { x } } { \sqrt { \left( 5 x ^ { 2 } + 3 \right) } }$$

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

   \(x\)	- 0.25	0	0.25	0.5	0.75
   \(y\)	0.462		0.653		0.698

2. Use the trapezium rule,with all the values of \(y\) from the completed table,to find an approximate value for
   . $$\int _ { - 0.25 } ^ { 0.75 } \frac { 2 ^ { x } } { \sqrt { \left( 5 x ^ { 2 } + 3 \right) } } \mathrm { d } x$$

3. $$y = \sqrt { \left( 3 ^ { x } + x \right) }$$

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

   \(x\)	0	0.25	0.5	0.75	1
   \(y\)	1	1.251			2

2. Use the trapezium rule with all the values of \(y\) from your table to find an approximation for the value of $$\int _ { 0 } ^ { 1 } \sqrt { \left( 3 ^ { x } + x \right) } \mathrm { d } x$$ You must show clearly how you obtained your answer.
3. Explain how the trapezium rule could be used to obtain a more accurate estimate for the value of $$\int _ { 0 } ^ { 1 } \sqrt { \left( 3 ^ { x } + x \right) } d x$$
   \includegraphics[max width=\textwidth, alt={}]{0aafa21b-25f4-4f36-b914-bbaf6cae7a66-10_2673_1948_107_118}

3. \(y = \sqrt { } \left( 10 x - x ^ { 2 } \right)\).

1. Complete the table below, giving the values of \(y\) to 2 decimal places.

   \(x\)	1	1.4	1.8	2.2	2.6	3
   \(y\)	3	3.47			4.39

2. Use the trapezium rule, with all the values of \(y\) from your table, to find an approximation for the value of \(\int _ { 1 } ^ { 3 } \sqrt { } \left( 10 x - x ^ { 2 } \right) \mathrm { d } x\).

6. $$y = \frac { 5 } { 3 x ^ { 2 } - 2 }$$

1. Complete the table below, giving the values of \(y\) to 2 decimal places.

   \(x\)	2	2.25	2.5	2.75	3
   \(y\)	0.5	0.38			0.2

2. Use the trapezium rule, with all the values of \(y\) from your table, to find an approximate value for \(\int _ { 2 } ^ { 3 } \frac { 5 } { 3 x ^ { 2 } - 2 } \mathrm {~d} x\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{be8f9187-055a-476f-974d-22e8e16e9996-08_537_743_941_603} \captionsetup{labelformat=empty} \caption{Figure 2}
   \end{figure} Figure 2 shows a sketch of part of the curve with equation \(y = \frac { 5 } { 3 x ^ { 2 } - 2 } , x > 1\).
   At the points \(A\) and \(B\) on the curve, \(x = 2\) and \(x = 3\) respectively.
   The region \(S\) is bounded by the curve, the straight line through \(B\) and ( 2,0 ), and the line through \(A\) parallel to the \(y\)-axis. The region \(S\) is shown shaded in Figure 2.
3. Use your answer to part (b) to find an approximate value for the area of \(S\).

5. The curve \(C\) has equation $$y = x \sqrt { } \left( x ^ { 3 } + 1 \right) , \quad 0 \leqslant x \leqslant 2$$

1. Complete the table below, giving the values of \(y\) to 3 decimal places at \(x = 1\) and \(x = 1.5\).

   \(x\)	0	0.5	1	1.5	2
   \(y\)	0	0.530			6

2. Use the trapezium rule, with all the \(y\) values from your table, to find an approximation for the value of \(\int _ { 0 } ^ { 2 } x \sqrt { } \left( x ^ { 3 } + 1 \right) \mathrm { d } x\), giving your answer to 3 significant figures. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{22ebc302-765c-4734-b312-b286ccb20be9-06_1110_644_1119_648} \captionsetup{labelformat=empty} \caption{Figure 2}
   \end{figure} Figure 2 shows the curve \(C\) with equation \(y = x \sqrt { } \left( x ^ { 3 } + 1 \right) , 0 \leqslant x \leqslant 2\), and the straight line segment \(l\), which joins the origin and the point \(( 2,6 )\). The finite region \(R\) is bounded by \(C\) and \(l\).
3. Use your answer to part (b) to find an approximation for the area of \(R\), giving your answer to 3 significant figures.
   (3) \section*{LU}

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

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

   \(x\)	0	0.5	1	1.5	2
   \(y\)			2.646	3.630

2. Use the trapezium rule, with all the values of \(y\) from your table, to find an approximation for the value of \(\int _ { 0 } ^ { 2 } \sqrt { } \left( 5 ^ { x } + 2 \right) \mathrm { d } x\).

1. $$y = 3 ^ { x } + 2 x$$

1. Complete the table below, giving the values of \(y\) to 2 decimal places.

   \(x\)	0	0.2	0.4	0.6	0.8	1
   \(y\)	1	1.65				5

2. Use the trapezium rule, with all the values of \(y\) from your table, to find an approximate value for \(\int _ { 0 } ^ { 1 } \left( 3 ^ { x } + 2 x \right) d x\).

7. $$y = \sqrt { } \left( 3 ^ { x } + x \right)$$

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

   \(x\)	0	0.25	0.5	0.75	1
   \(y\)	1	1.251			2

2. Use the trapezium rule with all the values of \(y\) from your table to find an approximation for the value of \(\int _ { 0 } ^ { 1 } \sqrt { } \left( 3 ^ { x } + x \right) \mathrm { d } x\) You must show clearly how you obtained your answer.

2. $$y = \frac { x } { \sqrt { ( 1 + x ) } }$$

1. Complete the table below with the value of \(y\) corresponding to \(x = 1.3\), giving your answer to 4 decimal places.

   \(x\)	1	1.1	1.2	1.3	1.4	1.5
   \(y\)	0.7071	0.7591	0.8090		0.9037	0.9487

2. Use the trapezium rule, with all the values of \(y\) in the completed table, to obtain an approximate value for $$\int _ { 1 } ^ { 1.5 } \frac { x } { \sqrt { } ( 1 + x ) } \mathrm { d } x$$ giving your answer to 3 decimal places.
   You must show clearly each stage of your working.

3. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the curve with equation \(y = \sqrt { } ( 2 x - 1 ) , x \geqslant 0.5\) The finite region \(R\), shown shaded in Figure 1, is bounded by the curve, the \(x\)-axis and the lines with equations \(x = 2\) and \(x = 10\). The table below shows corresponding values of \(x\) and \(y\) for \(y = \sqrt { } ( 2 x - 1 )\).

1. Complete the table with the values of \(y\) corresponding to \(x = 4\) and \(x = 8\).
2. Use the trapezium rule, with all the values of \(y\) in the completed table, to find an approximate value for the area of \(R\), giving your answer to 2 decimal places.
3. State whether your approximate value in part (b) is an overestimate or an underestimate for the area of \(R\).

1. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the curve with equation \(y = \sqrt { } \left( x ^ { 2 } + 1 \right) , x \geqslant 0\) The finite region \(R\), shown shaded in Figure 1, is bounded by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 2\) The table below shows corresponding values for \(x\) and \(y\) for \(y = \sqrt { } \left( x ^ { 2 } + 1 \right)\).

1. Complete the table above, giving the missing value of \(y\) to 3 decimal places.
2. Use the trapezium rule, with all the values of \(y\) in the completed table, to find an approximate value for the area of \(R\), giving your answer to 2 decimal places.

2. The curve \(C\) has equation $$y = 8 - 2 ^ { x - 1 } , \quad 0 \leqslant x \leqslant 4$$

1. Complete the table below with the value of \(y\) corresponding to \(x = 1\)

   \(x\)	0	1	2	3	4
   \(y\)	7.5		6	4	0

2. Use the trapezium rule, with all the values of \(y\) in the completed table, to find an approximate value for \(\int _ { 0 } ^ { 4 } \left( 8 - 2 ^ { x - 1 } \right) \mathrm { d } x\) \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{582cda45-80fc-43a8-90e6-1cae08cb1534-03_650_606_1016_671} \captionsetup{labelformat=empty} \caption{Figure 1}
   \end{figure} Figure 1 shows a sketch of the curve \(C\) with equation \(y = 8 - 2 ^ { x - 1 } , \quad 0 \leqslant x \leqslant 4\) The curve \(C\) meets the \(x\)-axis at the point \(A\) and meets the \(y\)-axis at the point \(B\).
   The region \(R\), shown shaded in Figure 1, is bounded by the curve \(C\) and the straight line through \(A\) and \(B\).
3. Use your answer to part (b) to find an approximate value for the area of \(R\).

1. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the curve with equation $$y = \frac { ( x + 2 ) ^ { \frac { 3 } { 2 } } } { 4 } , \quad x \geqslant - 2$$ The finite region \(R\), shown shaded in Figure 1, is bounded by the curve, the \(x\)-axis and the line with equation \(x = 10\) The table below shows corresponding values of \(x\) and \(y\) for \(y = \frac { ( x + 2 ) ^ { \frac { 3 } { 2 } } } { 4 }\)

1. Complete the table, giving values of \(y\) corresponding to \(x = 2\) and \(x = 6\)

   \(x\)	- 2	2	6	10
   \(y\)	0			\(6 \sqrt { } 3\)

2. Use the trapezium rule, with all the values of \(y\) from the completed table, to find an approximate value for the area of \(R\), giving your answer to 3 decimal places.

5. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the curve \(C\) with equation $$y = x \cos x , \quad x \in \mathbb { R }$$ The finite region \(R\), shown shaded in Figure 1, is bounded by the curve \(C\) and the \(x\)-axis for \(\frac { 3 \pi } { 2 } \leqslant x \leqslant \frac { 5 \pi } { 2 }\)

1. Complete the table below with the exact value of \(y\) corresponding to \(x = \frac { 7 \pi } { 4 }\) and with the exact value of \(y\) corresponding to \(x = \frac { 9 \pi } { 4 }\)

   \(x\)	\(\frac { 3 \pi } { 2 }\)	\(\frac { 7 \pi } { 4 }\)	\(2 \pi\)	\(\frac { 9 \pi } { 4 }\)	\(\frac { 5 \pi } { 2 }\)
   \(y\)	0		\(2 \pi\)		0

2. Use the trapezium rule, with all five \(y\) values in the completed table, to find an approximate value for the area of \(R\), giving your answer to 4 significant figures.
3. Find $$\int x \cos x d x$$
4. Using your answer from part (c), find the exact area of the region \(R\).

13. \begin{figure}[h] \end{figure} Figure 5 shows a sketch of part of the curve with equation \(y = 2 - \ln x , x > 0\) The finite region \(R\), shown shaded in Figure 5, is bounded by the curve, the \(x\)-axis and the line with equation \(x = \mathrm { e }\). The table below shows corresponding values of \(x\) and \(y\) for \(y = 2 - \ln x\)

1. Complete the table giving the value of \(y\) to 4 decimal places.
2. Use the trapezium rule, with all the values of \(y\) in the completed table, to obtain an estimate for the area of \(R\), giving your answer to 3 decimal places.
3. Use integration by parts to show that \(\int ( \ln x ) ^ { 2 } \mathrm {~d} x = x ( \ln x ) ^ { 2 } - 2 x \ln x + 2 x + c\) The area \(R\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
4. Use calculus to find the exact volume of the solid generated. Write your answer in the form \(\pi \mathrm { e } ( p \mathrm { e } + q )\), where \(p\) and \(q\) are integers to be found.

6. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve \(C\) with equation \(y = 2 \mathrm { e } ^ { - x } \sqrt { \sin x } , 0 \leqslant x \leqslant \pi\). The finite region \(R\), shown shaded in Figure 1, is bounded by the curve and the \(x\)-axis.

1. Complete the table below with the value of \(y\) corresponding to \(x = \frac { \pi } { 2 }\), giving your answer to 5 decimal places.

   \(x\)	0	\(\frac { \pi } { 4 }\)	\(\frac { \pi } { 2 }\)	\(\frac { 3 \pi } { 4 }\)	\(\pi\)
   \(y\)	0	0.76679		0.15940	0

2. Use the trapezium rule, with all the values of \(y\) in the completed table, to obtain an estimate for the area of the region \(R\). Give your answer to 4 decimal places.
3. Given \(y = 2 \mathrm { e } ^ { - x } \sqrt { \sin x }\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) for \(0 < x < \pi\). The curve \(C\) has a maximum turning point when \(x = a\).
4. Use your answer to part (c) to find the value of \(a\), giving your answer to 3 decimal places.

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

1. Complete the table below with the values of \(y\) corresponding to \(x = \frac { \pi } { 4 }\) and \(x = \frac { \pi } { 2 }\), giving your answers to 5 decimal places.

   \(x\)	0	\(\frac { \pi } { 4 }\)	\(\frac { \pi } { 2 }\)	\(\frac { 3 \pi } { 4 }\)	\(\pi\)
   \(y\)	0			8.87207	0

2. Use the trapezium rule, with all the values of \(y\) in the completed table, to obtain an estimate for the area of the region \(R\). Give your answer to 4 decimal places. The curve \(y = \mathrm { e } ^ { x } \sqrt { \sin x } , 0 \leqslant x \leqslant \pi\), has a maximum turning point at \(Q\), shown in Figure 1.
3. Find the \(x\) coordinate of \(Q\).

1. \begin{figure}[h] \end{figure} The curve shown in Figure 1 has equation \(y = \mathrm { e } ^ { x } \sqrt { } ( \sin x ) , 0 \leqslant x \leqslant \pi\). The finite region \(R\) bounded by the curve and the \(x\)-axis is shown shaded in Figure 1.

1. Complete the table below with the values of \(y\) corresponding to \(x = \frac { \pi } { 4 }\) and \(\frac { \pi } { 2 }\), giving your answers to 5 decimal places.

   \(x\)	0	\(\frac { \pi } { 4 }\)	\(\frac { \pi } { 2 }\)	\(\frac { 3 \pi } { 4 }\)	\(\pi\)
   \(y\)	0			8.87207	0

2. Use the trapezium rule, with all the values in the completed table, to obtain an estimate for the area of the region \(R\). Give your answer to 4 decimal places.

6. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of the curve with equation \(y = \frac { 2 \sin 2 x } { ( 1 + \cos x ) } , 0 \leqslant x \leqslant \frac { \pi } { 2 }\).
The finite region \(R\), shown shaded in Figure 3, is bounded by the curve and the \(x\)-axis. The table below shows corresponding values of \(x\) and \(y\) for \(y = \frac { 2 \sin 2 x } { ( 1 + \cos x ) }\).

1. Complete the table above giving the missing value of \(y\) to 5 decimal places.
2. Use the trapezium rule, with all the values of \(y\) in the completed table, to obtain an estimate for the area of \(R\), giving your answer to 4 decimal places.
3. Using the substitution \(u = 1 + \cos x\), or otherwise, show that $$\int \frac { 2 \sin 2 x } { ( 1 + \cos x ) } d x = 4 \ln ( 1 + \cos x ) - 4 \cos x + k$$ where \(k\) is a constant.
4. Hence calculate the error of the estimate in part (b), giving your answer to 2 significant figures.

4. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the curve with equation \(y = \frac { 4 \mathrm { e } ^ { - x } } { 3 \sqrt { } \left( 1 + 3 \mathrm { e } ^ { - x } \right) }\) The finite region \(R\), shown shaded in Figure 1, is bounded by the curve, the \(x\)-axis, the line \(x = - 3 \ln 2\) and the \(y\)-axis. The table below shows corresponding values of \(x\) and \(y\) for \(y = \frac { 4 \mathrm { e } ^ { - x } } { 3 \sqrt { } \left( 1 + 3 \mathrm { e } ^ { - x } \right) }\)

1. Complete the table above by giving the missing value of \(y\) to 4 decimal places.
2. Use the trapezium rule, with all the values of \(y\) in the completed table, to obtain an estimate for the area of \(R\), giving your answer to 2 decimal places.
3. 1. Using the substitution \(u = 1 + 3 \mathrm { e } ^ { - x }\), or otherwise, find $$\int \frac { 4 \mathrm { e } ^ { - x } } { 3 \sqrt { } \left( 1 + 3 \mathrm { e } ^ { - x } \right) } \mathrm { d } x$$
   2. Hence find the value of the area of \(R\).

1. \begin{figure}[h] \end{figure} Figure 1 shows part of the curve with equation \(y = \mathrm { e } ^ { 0.5 x ^ { 2 } }\). The finite region \(R\), shown shaded in Figure 1, is bounded by the curve, the \(x\)-axis, the \(y\)-axis and the line \(x = 2\).

1. Complete the table with the values of \(y\) corresponding to \(x = 0.8\) and \(x = 1.6\).

   \(x\)	0	0.4	0.8	1.2	1.6	2
   \(y\)	\(\mathrm { e } ^ { 0 }\)	\(\mathrm { e } ^ { 0.08 }\)		\(\mathrm { e } ^ { 0.72 }\)		\(\mathrm { e } ^ { 2 }\)

2. Use the trapezium rule with all the values in the table to find an approximate value for the area of \(R\), giving your answer to 4 significant figures.

2. \begin{figure}[h] \end{figure} Figure 1 shows the finite region \(R\) bounded by the \(x\)-axis, the \(y\)-axis and the curve with equation \(y = 3 \cos \left( \frac { x } { 3 } \right) , 0 \leqslant x \leqslant \frac { 3 \pi } { 2 }\).
The table shows corresponding values of \(x\) and \(y\) for \(y = 3 \cos \left( \frac { x } { 3 } \right)\).

1. Complete the table above giving the missing value of \(y\) to 5 decimal places.
2. Using the trapezium rule, with all the values of \(y\) from the completed table, find an approximation for the area of \(R\), giving your answer to 3 decimal places.
3. Use integration to find the exact area of \(R\).