Deduce related integral value

2 \includegraphics[max width=\textwidth, alt={}, center]{2c27f384-5289-4c1b-9199-6b4c6ac81e38-2_645_750_429_699} The diagram shows the curve \(y = \sqrt { } \left( 1 + x ^ { 3 } \right)\). Region \(A\) is bounded by the curve and the lines \(x = 0\), \(x = 2\) and \(y = 0\). Region \(B\) is bounded by the curve and the lines \(x = 0\) and \(y = 3\).

1. Use the trapezium rule with two intervals to find an approximation to the area of region \(A\). Give your answer correct to 2 decimal places.
2. Deduce an approximation to the area of region \(B\) and explain why this approximation underestimates the true area of region \(B\).

2 \includegraphics[max width=\textwidth, alt={}, center]{ee420db2-bef4-4c2b-8dd2-c8f439dd561e-2_645_750_429_699} The diagram shows the curve \(y = \sqrt { } \left( 1 + x ^ { 3 } \right)\). Region \(A\) is bounded by the curve and the lines \(x = 0\), \(x = 2\) and \(y = 0\). Region \(B\) is bounded by the curve and the lines \(x = 0\) and \(y = 3\).

1. Use the trapezium rule with two intervals to find an approximation to the area of region \(A\). Give your answer correct to 2 decimal places.
2. Deduce an approximation to the area of region \(B\) and explain why this approximation underestimates the true area of region \(B\).

1. (a) Given that \(a\) is a constant, \(a > 1\), sketch the graph of

$$y = a ^ { x } , \quad x \in \mathbb { R }$$ On your diagram show the coordinates of the point where the graph crosses the \(y\)-axis.
(2) The table below shows corresponding values of \(x\) and \(y\) for \(y = 2 ^ { x }\) (b) Use the trapezium rule, with all of the values of \(y\) from the table, to find an approximate value, to 2 decimal places, for $$\int _ { - 4 } ^ { 4 } 2 ^ { x } \mathrm {~d} x$$ (c) Use the answer to part (b) to find an approximate value for

1. \(\int _ { - 4 } ^ { 4 } 2 ^ { x + 2 } \mathrm {~d} x\)
2. \(\int _ { - 4 } ^ { 4 } \left( 3 + 2 ^ { x } \right) \mathrm { d } x\)
   \includegraphics[max width=\textwidth, alt={}, center]{bb1becd5-96c1-426d-9b85-4bbc4a61af27-23_86_47_2617_1886}

1. The table below shows corresponding values of \(x\) and \(y\) for \(y = \log _ { 2 } ( 2 x )\)

The values of \(y\) are given to 2 decimal places as appropriate. Using the trapezium rule with all the values of \(y\) in the given table,

1. obtain an estimate for \(\int _ { 2 } ^ { 14 } \log _ { 2 } ( 2 x ) \mathrm { d } x\), giving your answer to one decimal place. Using your answer to part (a) and making your method clear, estimate
2. 1. \(\quad \int _ { 2 } ^ { 14 } \frac { \log _ { 2 } \left( 4 x ^ { 2 } \right) } { 5 } \mathrm {~d} x\)
   2. \(\int _ { 2 } ^ { 14 } \log _ { 2 } \left( \frac { 2 } { x } \right) \mathrm { d } x\)

      \(x\)	2	5	8	11	14
      \(y\)	2	3.32	4	4.46	4.81

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

$$y = 2 ^ { 5 - \sqrt { x } }$$ The values of \(y\) are given to 3 decimal places. Using the trapezium rule with all the values of \(y\) in the given table,

1. obtain an estimate for $$\int _ { 5 } ^ { 7 } 2 ^ { 5 - \sqrt { x } } \mathrm {~d} x$$ giving your answer to 2 decimal places.
2. Using your answer to part (a) and making your method clear, estimate
   1. \(\quad \int _ { 5 } ^ { 7 } 2 ^ { 6 - \sqrt { x } } \mathrm {~d} x\)
   2. \(\int _ { 5 } ^ { 7 } \left( 3 + 2 ^ { 5 - \sqrt { x } } \right) \mathrm { d } x\)

1. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\) The table below shows some corresponding values of \(x\) and \(y\) for this curve.
The values of \(y\) are given to 3 decimal places. Using the trapezium rule with all the values of \(y\) in the given table,

1. obtain an estimate for $$\int _ { - 1 } ^ { 1 } \mathrm { f } ( x ) \mathrm { d } x$$ giving your answer to 2 decimal places.
2. Use your answer to part (a) to estimate
   1. \(\int _ { - 1 } ^ { 1 } ( \mathrm { f } ( x ) - 2 ) \mathrm { d } x\)
   2. \(\int _ { 1 } ^ { 3 } \mathrm { f } ( x - 2 ) \mathrm { d } x\)

1. (a) Sketch the curve with equation

$$y = a ^ { - x } + 4$$ where \(a\) is a constant and \(a > 1\) On your sketch show

• the coordinates of the point of intersection of the curve with the \(y\)-axis
• the equation of the asymptote to the curve.

The table above shows corresponding values of \(x\) and \(y\) for \(y = 3 ^ { - \frac { 1 } { 2 } x } + 4\) The values of \(y\) are given to four significant figures, as appropriate.
Using the trapezium rule with all the values of \(y\) in the table,
(b) find an approximate value for $$\int _ { - 4 } ^ { 8.5 } \left( 3 ^ { - \frac { 1 } { 2 } x } + 4 \right) d x$$ giving your answer to two significant figures.
(c) Using the answer to part (b), find an approximate value for

1. \(\int _ { - 4 } ^ { 8.5 } \left( 3 ^ { - \frac { 1 } { 2 } x } \right) \mathrm { d } x\)
2. \(\int _ { - 4 } ^ { 8.5 } \left( 3 ^ { - \frac { 1 } { 2 } x } + 4 \right) \mathrm { d } x + \int _ { - 8.5 } ^ { 4 } \left( 3 ^ { \frac { 1 } { 2 } x } + 4 \right) \mathrm { d } x\)

2. \begin{figure}[h] \end{figure} Figure 1 shows the graph of $$y = 1 - \log _ { 10 } ( \sin x ) \quad 0 < x < \pi$$ where \(x\) is in radians. The table below shows some values of \(x\) and \(y\) for this graph, with values of \(y\) given to 3 decimal places.

1. Complete the table above, giving values of \(y\) to 3 decimal places.
2. Use the trapezium rule with all the \(y\) values in the completed table to find, to 2 decimal places, an estimate for $$\int _ { 0.5 } ^ { 3 } \left( 1 - \log _ { 10 } ( \sin x ) \right) \mathrm { d } x$$
3. Use your answer to part (b) to find an estimate for $$\int _ { 0.5 } ^ { 3 } \left( 3 + \log _ { 10 } ( \sin x ) \right) \mathrm { d } x$$

1. (a) Sketch the curve with equation

$$y = a ^ { x } + 4$$ where \(a\) is a positive constant greater than 1
On your sketch, show

• the coordinates of the point of intersection of the curve with the \(y\)-axis
• the equation of the asymptote of the curve

The table shows corresponding values of \(x\) and \(y\) for $$y = 2 ^ { x } - 2 x$$ with the values of \(y\) given to 4 decimal places as appropriate.
Using the trapezium rule with all the values of \(y\) in the given table,
(b) obtain an estimate for \(\int _ { 2 } ^ { 3.5 } \left( 2 ^ { x } - 2 x \right) \mathrm { d } x\), giving your answer to 2 decimal places.
(c) Using your answer to part (b) and making your method clear, estimate

1. \(\int _ { 2 } ^ { 3.5 } \left( 2 ^ { x } + 2 x \right) \mathrm { d } x\)
2. \(\int _ { 2 } ^ { 3.5 } \left( 2 ^ { x + 1 } - 4 x \right) \mathrm { d } x\)

5. (a) Given \(0 < a < 1\), sketch the curve with equation $$y = a ^ { x }$$ showing the coordinates of the point at which the curve crosses the \(y\)-axis. The table above shows corresponding values of \(x\) and \(y\) for \(y = x ^ { 2 } + \left( \frac { 1 } { 2 } \right) ^ { x }\) The values of \(y\) are given to 4 significant figures as appropriate.
Using the trapezium rule with all the values of \(y\) in the given table,
(b) obtain an estimate for \(\int _ { 2 } ^ { 4 } \left( x ^ { 2 } + \left( \frac { 1 } { 2 } \right) ^ { x } \right) \mathrm { d } x\) Using your answer to part (b) and making your method clear, estimate
(c) \(\quad \int _ { 2 } ^ { 4 } \left( x ( x - 3 ) + \left( \frac { 1 } { 2 } \right) ^ { x } \right) \mathrm { d } x\)

3. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the curve with equation \(y = \log _ { 10 } x\) The region \(R\), shown shaded in Figure 1, is bounded by the curve, the line with equation \(x = 2\), the \(x\)-axis and the line with equation \(x = 14\) Using the trapezium rule with four strips of equal width,

1. show that the area of \(R\) is approximately 10.10
2. Explain how the trapezium rule could be used to obtain a more accurate estimate for the area of \(R\).
3. Using the answer to part (a) and making your method clear, estimate the value of
   1. \(\quad \int _ { 2 } ^ { 14 } \log _ { 10 } \sqrt { x } \mathrm {~d} x\)
   2. \(\int _ { 2 } ^ { 14 } \log _ { 10 } 100 x ^ { 3 } \mathrm {~d} x\)

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.

\begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the graph of \(y = \frac { 12 } { \sqrt { \left( x ^ { 2 } - 2 \right) } } , x \geqslant 2\) The table below gives values of \(y\) rounded to 3 decimal places.

1. Use the trapezium rule with all the values of \(y\) from the table to find an approximate value, to 2 decimal places, for $$\int _ { 2 } ^ { 11 } \frac { 12 } { \sqrt { \left( x ^ { 2 } - 2 \right) } } \mathrm { d } x$$
2. Use your answer to part (a) to estimate a value for $$\int _ { 2 } ^ { 11 } \left( 1 + \frac { 6 } { \sqrt { \left( x ^ { 2 } - 2 \right) } } \right) d x$$

6. (i) Write down the exact value of \(\cos \frac { \pi } { 6 }\). The finite region \(R\) is bounded by the curve \(y = \cos ^ { 2 } x\), where \(x\) is measured in radians, the positive coordinate axes and the line \(x = \frac { \pi } { 3 }\).
(ii) Use the trapezium rule with two intervals of equal width to estimate the area of \(R\), giving your answer to 3 significant figures. The finite region \(S\) is bounded by the curve \(y = \sin ^ { 2 } x\), where \(x\) is measured in radians, the positive coordinate axes and the line \(x = \frac { \pi } { 3 }\).
(iii) Using your answer to part (b), find an estimate for the area of \(S\).

8 \includegraphics[max width=\textwidth, alt={}, center]{d858728a-3371-4755-880c-54f96c5e5156-4_787_742_276_719} The diagram shows part of the curve \(y = \ln \left( 5 - x ^ { 2 } \right)\) which meets the \(x\)-axis at the point \(P\) with coordinates \(( 2,0 )\). The tangent to the curve at \(P\) meets the \(y\)-axis at the point \(Q\). The region \(A\) is bounded by the curve and the lines \(x = 0\) and \(y = 0\). The region \(B\) is bounded by the curve and the lines \(P Q\) and \(x = 0\).

1. Find the equation of the tangent to the curve at \(P\).
2. Use Simpson's Rule with four strips to find an approximation to the area of the region \(A\), giving your answer correct to 3 significant figures.
3. Deduce an approximation to the area of the region \(B\).

8 \includegraphics[max width=\textwidth, alt={}, center]{1216a06e-7e14-48d7-a7ca-7acd8d71af5f-4_538_1443_262_351} The diagram shows the curve with equation \(y = x ^ { 8 } \mathrm { e } ^ { - x ^ { 2 } }\). The curve has maximum points at \(P\) and \(Q\). The shaded region \(A\) is bounded by the curve, the line \(y = 0\) and the line through \(Q\) parallel to the \(y\)-axis. The shaded region \(B\) is bounded by the curve and the line \(P Q\).

1. Show by differentiation that the \(x\)-coordinate of \(Q\) is 2 .
2. Use Simpson's rule with 4 strips to find an approximation to the area of region \(A\). Give your answer correct to 3 decimal places.
3. Deduce an approximation to the area of region \(B\).

8 The definite integral \(I\) is defined by $$I = \int _ { 0 } ^ { 6 } 2 ^ { x } \mathrm {~d} x$$

1. Use Simpson's rule with 6 strips to find an approximate value of \(I\).
2. By first writing \(2 ^ { x }\) in the form \(\mathrm { e } ^ { k x }\), where the constant \(k\) is to be determined, find the exact value of \(I\).
3. Use the answers to parts (i) and (ii) to deduce that \(\ln 2 \approx \frac { 9 } { 13 }\).

4

1. Use the trapezium rule, with 4 strips each of width 0.5 , to find an approximate value for $$\int _ { 3 } ^ { 5 } \log _ { 10 } ( 2 + x ) d x$$ giving your answer correct to 3 significant figures.
2. Use your answer to part (i) to deduce an approximate value for \(\int _ { 3 } ^ { 5 } \log _ { 10 } \sqrt { 2 + x } \mathrm {~d} x\), showing your method clearly.

9 \includegraphics[max width=\textwidth, alt={}, center]{9e95415c-00f5-4b52-a443-0b946602b3b4-4_387_624_287_717} The diagram shows part of the curve \(y = - 3 + 2 \sqrt { x + 4 }\). The point \(P ( 5,3 )\) lies on the curve. Region \(A\) is bounded by the curve, the \(x\)-axis, the \(y\)-axis and the line \(x = 5\). Region \(B\) is bounded by the curve, the \(y\)-axis and the line \(y = 3\).

1. Use the trapezium rule, with 2 strips each of width 2.5 , to find an approximate value for the area of region \(A\), giving your answer correct to 3 significant figures.
2. Use your answer to part (i) to deduce an approximate value for the area of region \(B\).
3. By first writing the equation of the curve in the form \(x = \mathrm { f } ( y )\), use integration to show that the exact area of region \(B\) is \(\frac { 14 } { 3 }\). \section*{END OF QUESTION PAPER} \section*{OCR \(^ { \text {N } }\)}

2

1. Use Simpson's rule with four strips to find an approximation to $$\int _ { 4 } ^ { 12 } \ln x \mathrm {~d} x$$ giving your answer correct to 2 decimal places.
2. Deduce an approximation to \(\int _ { 4 } ^ { 12 } \ln \left( x ^ { 10 } \right) \mathrm { d } x\).

6 The value of \(\int _ { 0 } ^ { 8 } \ln \left( 3 + x ^ { 2 } \right) \mathrm { d } x\) obtained by using Simpson's rule with four strips is denoted by \(A\).

1. Find the value of \(A\) correct to 3 significant figures.
2. Explain why an approximate value of \(\int _ { 0 } ^ { 8 } \ln \left( 9 + 6 x ^ { 2 } + x ^ { 4 } \right) \mathrm { d } x\) is \(2 A\).
3. Explain why an approximate value of \(\int _ { 0 } ^ { 8 } \ln \left( 3 \mathrm { e } + \mathrm { e } x ^ { 2 } \right) \mathrm { d } x\) is \(A + 8\).

3

1. Use Simpson's rule with four strips to find an approximation to $$\int _ { 0 } ^ { 2 } \mathrm { e } ^ { \sqrt { x } } \mathrm {~d} x$$ giving your answer correct to 3 significant figures.
2. Deduce an approximation to \(\int _ { 0 } ^ { 2 } \left( 1 + 10 \mathrm { e } ^ { \sqrt { x } } \right) \mathrm { d } x\).

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. \(\int _ { 0 } ^ { 0.5 } \sqrt { \left( 2 ^ { 2 x } + 2 x \right) } \mathrm { d } x\)

1. The table below shows corresponding values of \(x\) and \(y\) for \(y = \log _ { 3 } ( x )\) The values of \(y\) are given to 2 decimal places as appropriate.

a. Obtain an estimate for \(\int _ { 3 } ^ { 9 } \log _ { 3 } ( x ) \mathrm { d } x\), giving your answer to two decimal places. Use your answer to part (a) and making your method clear, estimate
b. i) \(\int _ { 3 } ^ { 9 } \log _ { 3 } \sqrt { x } \mathrm {~d} x\) ii) \(\int _ { 3 } ^ { 18 } \log _ { 3 } \left( 9 x ^ { 3 } \right) \mathrm { d } x\)