1.09f Trapezium rule: numerical integration

3. \begin{figure}[h] \end{figure} Figure 1 shows the curve with equation \(y = \frac { 4 x } { ( x + 1 ) ^ { 2 } }\).
The shaded region is bounded by the curve, the \(x\)-axis and the line \(x = 1\).

1. Use the trapezium rule with four intervals of equal width to find an estimate for the area of the shaded region.
2. State, with a reason, whether your answer to part (a) is an under-estimate or an over-estimate of the true area.

2. \begin{figure}[h] \end{figure} Figure 1 shows the curve with equation \(y = \sqrt { 4 x - 1 }\). Use the trapezium rule with five equally-spaced ordinates to estimate the area of the shaded region bounded by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 3\).

1 Use the mid-ordinate rule with four strips of equal width to find an estimate for \(\int _ { 1 } ^ { 5 } \frac { 1 } { 1 + \ln x } \mathrm {~d} x\), giving your answer to three significant figures.
(4 marks)

6

1. Sketch the curve with equation \(y = \operatorname { cosec } x\) for \(0 < x < \pi\).
2. Use the mid-ordinate rule with four strips to find an estimate for \(\int _ { 0.1 } ^ { 0.5 } \operatorname { cosec } x \mathrm {~d} x\), giving your answer to three significant figures.

6

1. Use the mid-ordinate rule with four strips to find an estimate for \(\int _ { 0 } ^ { 0.4 } \cos \sqrt { 3 x + 1 } \mathrm {~d} x\), giving your answer to three significant figures.
2. Use the substitution \(u = 3 x + 1\) to find the exact value of \(\int _ { 0 } ^ { 1 } x \sqrt { 3 x + 1 } \mathrm {~d} x\).
   (6 marks)

1

1. Use Simpson's rule with 7 ordinates (6 strips) to find an estimate for \(\int _ { 0 } ^ { 3 } 4 ^ { x } \mathrm {~d} x\).
2. A curve is defined by the equation \(y = 4 ^ { x }\). The curve intersects the line \(y = 8 - 2 x\) at a single point where \(x = \alpha\).
   1. Show that \(\alpha\) lies between 1.2 and 1.3.
   2. The equation \(4 ^ { x } = 8 - 2 x\) can be rearranged into the form \(x = \frac { \ln ( 8 - 2 x ) } { \ln 4 }\). Use the iterative formula \(x _ { n + 1 } = \frac { \ln \left( 8 - 2 x _ { n } \right) } { \ln 4 }\) with \(x _ { 1 } = 1.2\) to find the values of \(x _ { 2 }\) and \(x _ { 3 }\), giving your answers to three decimal places.
      (2 marks)

2

1. Use Simpson's rule, with five ordinates (four strips), to calculate an estimate for $$\int _ { 0 } ^ { 4 } \frac { x } { x ^ { 2 } + 2 } \mathrm {~d} x$$ Give your answer to four significant figures.
2. Show that the exact value of \(\int _ { 0 } ^ { 4 } \frac { x } { x ^ { 2 } + 2 } \mathrm {~d} x\) is \(\ln k\), where \(k\) is an integer. (5 marks)

8 The diagram shows part of the graph of \(y = \mathrm { e } ^ { 2 x } + 3\). \includegraphics[max width=\textwidth, alt={}, center]{d5b78fa6-ea3c-497b-94d8-1d5f61288aa5-4_833_1034_1027_513}

1. Describe a sequence of two geometrical transformations that maps the graph of \(y = \mathrm { e } ^ { x }\) onto the graph of \(y = \mathrm { e } ^ { 2 x } + 3\).
2. Use the mid-ordinate rule with four strips of equal width to find an estimate for the area of the shaded region \(A\), giving your answer to three significant figures.
3. Find the exact value of the area of the shaded region \(A\).
4. The region \(B\) is indicated on the diagram. Find the area of the region \(B\), giving your answer in the form \(p \mathrm { e } ^ { 8 } + q \mathrm { e } ^ { 4 }\), where \(p\) and \(q\) are numbers to be determined.

6

1. Use the mid-ordinate rule with four strips to find an estimate for \(\int _ { 1 } ^ { 5 } \ln x \mathrm {~d} x\), giving your answer to three significant figures.
2. 1. Given that \(y = x \ln x\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
   2. Hence, or otherwise, find \(\int \ln x \mathrm {~d} x\).
   3. Find the exact value of \(\int _ { 1 } ^ { 5 } \ln x \mathrm {~d} x\).

6 The diagram shows the curve with equation \(y = \left( \mathrm { e } ^ { 3 x } + 1 \right) ^ { \frac { 1 } { 2 } }\) for \(x \geqslant 0\). \includegraphics[max width=\textwidth, alt={}, center]{6ce5aa0d-0a73-4bc4-aabc-314c0434e4f5-5_483_611_402_717}

1. Find the gradient of the curve \(y = \left( \mathrm { e } ^ { 3 x } + 1 \right) ^ { \frac { 1 } { 2 } }\) at the point where \(x = \ln 2\).
2. Use the mid-ordinate rule with four strips to find an estimate for \(\int _ { 0 } ^ { 2 } \left( \mathrm { e } ^ { 3 x } + 1 \right) ^ { \frac { 1 } { 2 } } \mathrm {~d} x\), giving your answer to three significant figures.
3. The shaded region \(R\) is bounded by the curve, the lines \(x = 0 , x = 2\) and the \(x\)-axis. Find the exact value of the volume of the solid generated when the region \(R\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.

6 The diagram shows the curve with equation \(y = \sqrt { 100 - 4 x ^ { 2 } }\), where \(x \geqslant 0\). \includegraphics[max width=\textwidth, alt={}, center]{a596af76-9680-4ccb-a512-5b2575414429-5_518_494_367_758}

1. Calculate the volume of the solid generated when the region bounded by the curve shown above and the coordinate axes is rotated through \(360 ^ { \circ }\) about the \(\boldsymbol { y }\)-axis, giving your answer in terms of \(\pi\).
2. Use the mid-ordinate rule with five strips of equal width to find an estimate for \(\int _ { 0 } ^ { 5 } \sqrt { 100 - 4 x ^ { 2 } } \mathrm {~d} x\), giving your answer to three significant figures.
3. The point \(P\) on the curve has coordinates \(( 3,8 )\).
   1. Find the gradient of the curve \(y = \sqrt { 100 - 4 x ^ { 2 } }\) at the point \(P\).
   2. Hence show that the equation of the tangent to the curve at the point \(P\) can be written as \(2 y + 3 x = 25\).
4. The shaded regions on the diagram below are bounded by the curve, the tangent at \(P\) and the coordinate axes. \includegraphics[max width=\textwidth, alt={}, center]{a596af76-9680-4ccb-a512-5b2575414429-5_642_546_1800_731} Use your answers to part (b) and part (c)(ii) to find an approximate value for the total area of the shaded regions. Give your answer to three significant figures.

4

1. Use Simpson's rule with 7 ordinates ( 6 strips) to find an approximation to \(\int _ { 0.5 } ^ { 2 } \frac { x } { 1 + x ^ { 3 } } \mathrm {~d} x\), giving your answer to three significant figures.
2. Find the exact value of \(\int _ { 0 } ^ { 1 } \frac { x ^ { 2 } } { 1 + x ^ { 3 } } \mathrm {~d} x\).

1 Use the mid-ordinate rule with four strips to find an estimate for \(\int _ { 0.4 } ^ { 1.2 } \cot \left( x ^ { 2 } \right) \mathrm { d } x\), giving your answer to three decimal places.

5 The diagram shows a sketch of the graph of \(y = \sqrt { 27 + x ^ { 3 } }\). \includegraphics[max width=\textwidth, alt={}, center]{063bbfa5-df49-44a1-8143-5e076397f63f-04_762_988_365_534}

1. The area of the shaded region, bounded by the curve, the \(x\)-axis and the lines \(x = 0\) and \(x = 4\), is given by \(\int _ { 0 } ^ { 4 } \sqrt { 27 + x ^ { 3 } } \mathrm {~d} x\). Use the mid-ordinate rule with five strips to find an estimate for this area. Give your answer to three significant figures.
2. With the aid of a diagram, explain whether the mid-ordinate rule applied in part (a) gives an estimate which is smaller than or greater than the area of the shaded region.
   (2 marks)

1 Use Simpson's rule, with five ordinates (four strips), to calculate an estimate for $$\int _ { 0 } ^ { \pi } x ^ { \frac { 1 } { 2 } } \sin x d x$$ Give your answer to four significant figures.
[0pt] [4 marks]

2 The curve with equation \(y = x ^ { x }\), where \(x > 0\), intersects the line \(y = 5\) at a single point, where \(x = \alpha\).

1. Show that \(\alpha\) lies between 2 and 3 .
2. Show that the equation \(x ^ { x } = 5\) can be rearranged into the form $$x = \mathrm { e } ^ { \left( \frac { \ln 5 } { x } \right) }$$
3. Use the iterative formula $$x _ { n + 1 } = \mathrm { e } ^ { \left( \frac { \ln 5 } { x _ { n } } \right) }$$ with \(x _ { 1 } = 2\) to find the values of \(x _ { 2 }\) and \(x _ { 3 }\), giving your answers to three decimal places.
4. 1. Use Simpson's rule with 7 ordinates ( 6 strips) to find an approximation to $$\int _ { 0.5 } ^ { 1.7 } \left( 5 - x ^ { x } \right) \mathrm { d } x$$ giving your answer to three significant figures.
   2. Hence find an approximation to \(\int _ { 0.5 } ^ { 1.7 } x ^ { x } \mathrm {~d} x\).

1. A measure of the effective voltage, \(M\) volts, in an electrical circuit is given by

$$M ^ { 2 } = \int _ { 0 } ^ { 1 } V ^ { 2 } \mathrm {~d} t$$ where \(V\) volts is the voltage at time \(t\) seconds. Pairs of values of \(V\) and \(t\) are given in the following table. Use the trapezium rule with five values of \(V ^ { 2 }\) to estimate the value of \(M\).
(6)

1. The following is a table of values for \(y = \sqrt { } ( 1 + \sin x )\), where \(x\) is in radians.

1. Find the value of \(p\) and the value of \(q\).
   (2)
2. Use the trapezium rule and all the values of \(y\) in the completed table to obtain an estimate of \(I\), where $$I = \int _ { 0 } ^ { 2 } \sqrt { } ( 1 + \sin x ) \mathrm { d } x$$ (4)
   

5. \begin{figure}[h] \end{figure} Figure 1 shows the curve with equation \(y = 4 x ^ { \frac { 1 } { 2 } } \mathrm { e } ^ { - x }\).
The shaded region is bounded by the curve, the \(x\)-axis and the line \(x = 2\).

1. Use the trapezium rule with four intervals of equal width to estimate the area of the shaded region. The shaded region is rotated through \(2 \pi\) radians about the \(x\)-axis.
2. Find, in terms of \(\pi\) and e, the exact volume of the solid formed.
   5. continued

8. \begin{figure}[h] \end{figure} Figure 2 shows the curve with equation \(y = \sqrt { \frac { x } { x + 1 } }\).
The shaded region is bounded by the curve, the \(x\)-axis and the line \(x = 3\).

1. 1. Use the trapezium rule with three strips to find an estimate for the area of the shaded region.
   2. Use the trapezium rule with six strips to find an improved estimate for the area of the shaded region. The shaded region is rotated through \(2 \pi\) radians about the \(x\)-axis.
2. Show that the volume of the solid formed is \(\pi ( 3 - \ln 4 )\).
   8. continued
   8. continued

4.

1. Use the trapezium rule with two intervals of equal width to find an estimate for the value of the integral $$\int _ { 0 } ^ { 3 } e ^ { \cos x } d x$$ giving your answer to 3 significant figures.
2. Use the trapezium rule with four intervals of equal width to find another estimate for the value of the integral to 3 significant figures.
3. Given that the true value of the integral lies between the estimates made in parts (a) and (b), comment on the shape of the curve \(y = \mathrm { e } ^ { \cos x }\) in the interval \(0 \leq x \leq 3\) and explain your answer.
   4. continued
   

3. \begin{figure}[h] \end{figure} Figure 1 shows the curve with equation \(y = \ln ( 2 + \cos x ) , 0 \leq x \leq \pi\).

1. Complete the table below for points on the curve, giving the \(y\) values to 4 decimal places.
2. Giving your answers to 3 decimal places, find estimates for the area of the region bounded by the curve and the coordinate axes using the trapezium rule with
   1. 1 strip,
   2. 2 strips,
   3. 4 strips.
3. Making your reasoning clear, suggest a value to 2 decimal places for the actual area of the region bounded by the curve and the coordinate axes.

   \(x\)	0	\(\frac { \pi } { 4 }\)	\(\frac { \pi } { 2 }\)	\(\frac { 3 \pi } { 4 }\)	\(\pi\)
   \(y\)	1.0986				0

   1. continued
   \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{3056ad22-f87b-46c3-86cf-d46939927465-06_563_983_146_379} \captionsetup{labelformat=empty} \caption{Figure 2}
   \end{figure} Figure 2 shows the curve with parametric equations $$x = \tan \theta , \quad y = \cos ^ { 2 } \theta , \quad - \frac { \pi } { 2 } < \theta < \frac { \pi } { 2 }$$ The shaded region bounded by the curve, the \(x\)-axis and the lines \(x = - 1\) and \(x = 1\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
   

1 A curve passes through the point (1,3) and satisfies the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { x } { 1 + x ^ { 3 } }$$ Starting at the point ( 1,3 ), use a step-by-step method with a step length of 0.1 to estimate the value of \(y\) at \(x = 1.2\). Give your answer to four decimal places.

1 A curve passes through the point \(( 2,3 )\) and satisfies the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { \sqrt { 2 + x } }$$ Starting at the point \(( 2,3 )\), use a step-by-step method with a step length of 0.5 to estimate the value of \(y\) at \(x = 3\). Give your answer to four decimal places.