1.09f Trapezium rule: numerical integration

The integral \(I\) is defined by $$I = \int_0^{13} (2x + 1)^{\frac{3}{2}} \, dx.$$

1. Use integration to find the exact value of \(I\). [4]
2. Use Simpson's rule with two strips to find an approximate value for \(I\). Give your answer correct to 3 significant figures. [3]

\includegraphics{figure_4} The diagram shows the curves \(y = \ln x\) and \(y = 2 \ln(x - 6)\). The curves meet at the point \(P\) which has \(x\)-coordinate \(a\). The shaded region is bounded by the curve \(y = 2 \ln(x - 6)\) and the lines \(x = a\) and \(y = 0\).

1. Give details of the pair of transformations which transforms the curve \(y = \ln x\) to the curve \(y = 2 \ln(x - 6)\). [3]
2. Solve an equation to find the value of \(a\). [4]
3. Use Simpson's rule with two strips to find an approximation to the area of the shaded region. [3]

\includegraphics{figure_7} The diagram shows the curve with equation \(y = 2x - e^{\frac{1}{2}x}\). The shaded region is bounded by the curve, the \(x\)-axis and the lines \(x = 2\) and \(x = 4\).

1. Find the area of the shaded region, giving your answer in terms of e. [4]

The shaded region is rotated through four right angles about the \(x\)-axis.

1. Using Simpson's rule with two strips, estimate the volume of the solid formed. [5]

\includegraphics{figure_2} Figure 2 shows the cross-section of a road tunnel and its concrete surround. The curved section of the tunnel is modelled by the curve with equation \(y = 8\sqrt{\sin \frac{\pi x}{10}}\), in the interval \(0 \leq x \leq 10\). The concrete surround is represented by the shaded area bounded by the curve, the \(x\)-axis and the lines \(x = -2\), \(x = 12\) and \(y = 10\). The units on both axes are metres.

1. Using this model, copy and complete the table below, giving the values of \(y\) to 2 decimal places.

   \(x\)	0	2	4	6	8	10
   \(y\)	0	6.13				0

   [2]

The area of the cross-section of the tunnel is given by \(\int_0^{10} y \, dx\).

1. Estimate this area, using the trapezium rule with all the values from your table. [4]
2. Deduce an estimate of the cross-sectional area of the concrete surround. [1]
3. State, with a reason, whether your answer in part (c) over-estimates or under-estimates the true value. [2]

\includegraphics{figure_2} Figure 2 shows part of the curve with equation $$y = e^x \cos x, \quad 0 \leq x \leq \frac{\pi}{2}.$$ The finite region \(R\) is bounded by the curve and the coordinate axes.

1. Calculate, to 2 decimal places, the \(y\)-coordinates of the points on the curve where \(x = 0\), \(\frac{\pi}{6}\), \(\frac{\pi}{3}\) and \(\frac{\pi}{2}\). [3]
2. Using the trapezium rule and all the values calculated in part (a), find an approximation for the area of \(R\). [4]
3. State, with a reason, whether your approximation underestimates or overestimates the area of \(R\). [2]

A student tests the accuracy of the trapezium rule by evaluating \(I\), where $$I = \int_{0.5}^{1.5} \left(\frac{3}{x} + x^4\right) dx.$$

1. Complete the student's table, giving values to 2 decimal places where appropriate.

   \(x\)	0.5	0.75	1	1.25	1.5
   \(\frac{3}{x} + x^4\)	6.06	4.32

   [2]
2. Use the trapezium rule, with all the values from your table, to calculate an estimate for the value of \(I\). [4]
3. Use integration to calculate the exact value of \(I\). [4]
4. Verify that the answer obtained by the trapezium rule is within 3\% of the exact value. [2]

Fig. 3 shows the curve \(y = x^3 + \sqrt{(\sin x)}\) for \(0 \leqslant x \leqslant \frac{\pi}{4}\). \includegraphics{figure_3}

1. Use the trapezium rule with 4 strips to estimate the area of the region bounded by the curve, the \(x\)-axis and the line \(x = \frac{\pi}{4}\), giving your answer to 3 decimal places. [4]
2. Suppose the number of strips in the trapezium rule is increased. Without doing further calculations, state, with a reason, whether the area estimate increases, decreases, or it is not possible to say. [1]

\includegraphics{figure_1} Figure 1 shows the curve with equation \(y = x\sqrt{\ln x}\), \(x \geq 1\). The shaded region is bounded by the curve, the \(x\)-axis and the line \(x = 3\).

1. Using the trapezium rule with two intervals of equal width, estimate the area of the shaded region. [4]

The shaded region is rotated through \(360°\) about the \(x\)-axis.

1. Find the exact volume of the solid formed. [7]

\includegraphics{figure_4} The diagram shows the curve \(y = e^{-\frac{1}{x}}\) for \(0 < x \leq 1\). A set of \((n - 1)\) rectangles is drawn under the curve as shown.

1. Explain why a lower bound for \(\int_0^1 e^{-\frac{1}{x}} dx\) can be expressed as $$\frac{1}{n}\left(e^{-n} + e^{-\frac{n}{2}} + e^{-\frac{n}{3}} + \ldots + e^{-\frac{n}{n-1}}\right).$$ [2]
2. Using a set of \(n\) rectangles, write down a similar expression for an upper bound for \(\int_0^1 e^{-\frac{1}{x}} dx\). [2]
3. Evaluate these bounds in the case \(n = 4\), giving your answers correct to 3 significant figures. [2]
4. When \(n > N\), the difference between the upper and lower bounds is less than 0.001. By expressing this difference in terms of \(n\), find the least possible value of \(N\). [3]

The graph of \(y = \frac{2x^3}{x^2 + 1}\) is shown for \(0 \leq x \leq 4\) Caroline is attempting to approximate the shaded area, A, under the curve using the trapezium rule by splitting the area into \(n\) trapezia.

1. When \(n = 4\)
   1. State the number of ordinates that Caroline uses. [1 mark]
   2. Calculate the area that Caroline should obtain using this method. Give your answer correct to two decimal places. [3 marks]
2. Show that the exact area of \(A\) is $$16 - \ln 17$$ Fully justify your answer. [5 marks]
3. Explain what would happen to Caroline's answer to part (a)(ii) as \(n \to \infty\) [1 mark]

Figure 2 below shows a 1.5 metre length of pipe. \includegraphics{figure_16} The symmetrical cross-section of the pipe is shown below, in Figure 3, where \(x\) and \(y\) are measured in centimetres. \includegraphics{figure_16_cross_section} Use the trapezium rule, with the values shown in the table below, to find the best estimate for the volume of the pipe. \begin{array}{|c|c|c|c|c|c|c|} \hline x & 0 & 0.4 & 0.8 & 1.2 & 1.6 & 2
\hline y & -3 & -2.943 & -2.752 & -2.353 & -1.572 & 0
\hline \end{array} [5 marks]

A particle is moving in a straight line with velocity \(v\text{ ms}^{-1}\) at time \(t\) seconds as shown by the graph below. \includegraphics{figure_15}

1. Use the trapezium rule with four strips to estimate the distance travelled by the particle during the time period \(20 \leq t \leq 100\) [4 marks]
2. Over the same time period, the curve can be very closely modelled by a particular quadratic. Explain how you could find an alternative estimate using this quadratic. [1 mark]

The trapezium rule is used to estimate the area of the shaded region in each of the graphs below. Identify the graph for which the trapezium rule produces an overestimate. Tick (\(\checkmark\)) one box. [1 mark] \includegraphics{figure_2}

The diagram shows part of the graph of \(y = e^{-x^2}\) \includegraphics{figure_7} The graph is formed from two convex sections, where the gradient is increasing, and one concave section, where the gradient is decreasing.

1. Find the values of \(x\) for which the graph is concave. [4 marks]
2. The finite region bounded by the \(x\)-axis and the lines \(x = 0.1\) and \(x = 0.5\) is shaded. \includegraphics{figure_7b} Use the trapezium rule, with 4 strips, to find an estimate for \(\int_{0.1}^{0.5} e^{-x^2} dx\) Give your estimate to four decimal places. [3 marks]
3. Explain with reference to your answer in part (a), why the answer you found in part (b) is an underestimate. [2 marks]
4. By considering the area of a rectangle, and using your answer to part (b), prove that the shaded area is 0.4 correct to 1 decimal place. [3 marks]

The function g is defined by $$g(x) = \mathrm{e}^{\sin x} \quad (0 \leq x \leq 2\pi)$$ The diagram below shows the graph of \(y = g(x)\) \includegraphics{figure_8}

1. Find the \(x\)-coordinate of each of the stationary points of the graph of \(y = g(x)\), giving your answers in exact form. [1 mark]
2. Use Simpson's rule with 3 ordinates to estimate $$\int_0^\pi g(x) \, \mathrm{d}x$$ giving your answer to two decimal places. [3 marks]
3. Explain how Simpson's rule could be used to find a more accurate estimate of the integral in part (b). [1 mark]

The diagram shows part of the graph of \(y = \cos^{-1} x\) \includegraphics{figure_7} The finite region enclosed by the graph of \(y = \cos^{-1} x\), the \(y\)-axis, the \(x\)-axis and the line \(x = 0.8\) is rotated by \(2\pi\) radians about the \(x\)-axis. Use Simpson's rule with five ordinates to estimate the volume of the solid formed. Give your answer to four decimal places. [5 marks]

The following algorithm produces a numerical approximation for the integral $$I = \int_A^B x^4 \, dx$$

Step 1	Start
Step 2	Input the values of A, B and N
Step 3	Let H = (B - A) / N
Step 4	Let C = H / 2
Step 5	Let D = 0
Step 6	Let D = D + A\(^4\) + B\(^4\)
Step 7	Let E = A
Step 8	Let E = E + H
Step 9	If E = B go to Step 12
Step 10	Let D = D + 2 × E\(^4\)
Step 11	Go to Step 8
Step 12	Let F = C × D
Step 13	Output F
Step 14	Stop

For the case when A = 1, B = 3 and N = 4,

1. 1. complete the table in the answer book to show the results obtained at each step of the algorithm.
   2. State the final output. [4]
2. Calculate, to 3 significant figures, the percentage error between the exact value of \(I\) and the value obtained from using the approximation to \(I\) in this case. [3]

The aerial view of a patio under construction is shown below. \includegraphics{figure_9} The curved edge of the patio is described by the equation \(9x^2 + 16y^2 = 144\), where \(x\) and \(y\) are measured in metres. To construct the patio, the area enclosed by the curve and the coordinate axes is to be covered with a layer of concrete of depth 0.06 m.

1. Show that the volume of concrete required for the construction of the patio is given by \(0.015 \int_0^4 \sqrt{144 - 9x^2}\,dx\). [3]
2. Use the trapezium rule with six ordinates to estimate the volume of concrete required. [4]
3. State whether your answer in part (b) is an overestimate or an underestimate of the volume required. Give a reason for your answer. [1]

\includegraphics{figure_3} Figure 3 shows a sketch of the curve with equation \(y = 2x^2 - x\). The finite region \(R\), shown shaded in Figure 3, is bounded by the curve, the line with equation \(x = -0.5\), the \(x\)-axis and the line with equation \(x = 1.5\).

1. The trapezium rule with four strips is used to find an estimate for the area of \(R\). Explain whether the estimate for R is an underestimate or overestimate to the true value for the area of \(R\). [1]

The estimate for R is found to be 2.58. Using this value, and showing your working,

1. estimate the value of \(\int_{-0.5}^{1.5} (2x^2 + 1 + 2x) \, dx\). [3]

\includegraphics{figure_1} Figure 1 shows part of the curve with equation \(y = e^{\frac{1}{5}x^2}\) for \(x \geq 0\) The finite region \(R\), shown shaded in Figure 1, is bounded by the curve, the \(y\)-axis, the \(x\)-axis, and the line with equation \(x = 2\) The table below shows corresponding values of \(x\) and \(y\) for \(y = e^{\frac{1}{5}x^2}\)

\(x\)	0	0.5	1	1.5	2
\(y\)	1	\(e^{0.05}\)	\(e^{0.2}\)	\(e^{0.45}\)	\(e^{0.8}\)

1. Use the trapezium rule, with all the values of \(y\) in the table, to find an estimate for the area of \(R\), giving your answer to 2 decimal places. [3]
2. Use your answer to part (a) to deduce an estimate for
   1. \(\int_0^2 \left( 4 + e^{\frac{1}{5}x^2} \right) dx\)
   2. \(\int_1^3 e^{\frac{1}{5}(x-1)^2} dx\) giving your answers to 2 decimal places. [2]