Trapezium rule with stated number of strips

2. \begin{figure}[h] \end{figure} Figure 1 shows the curve with equation \(y = 2 ^ { x }\).
Use the trapezium rule with four intervals of equal width to estimate the area of the shaded region bounded by the curve, the \(x\)-axis and the lines \(x = - 2\) and \(x = 2\).

6. \begin{figure}[h] \end{figure} Figure 1 shows the curve with equation \(y = 4 x + \frac { 1 } { x } , x > 0\).

1. Find the coordinates of the minimum point of the curve. The shaded region \(R\) is bounded by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 4\).
2. Use the trapezium rule with three intervals of equal width to estimate the area of \(R\).

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

1. Use the trapezium rule with four strips of equal width to estimate the area of \(R\). The cross-section of a support for a bookshelf is modelled by \(R\) with 1 unit on each axis representing 8 cm . Given that the support is 2 cm thick,
2. find an estimate for the volume of the support.

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)

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.

2 Use the trapezium rule with four ordinates (three strips) to find an approximate value for $$\int _ { 0 } ^ { 3 } \sqrt { 2 ^ { x } } \mathrm {~d} x$$ giving your answer to three decimal places.

2

1. Use the trapezium rule, with four strips each of width 0.25 , to find an approximate value for \(\int _ { 0 } ^ { 1 } \frac { 1 } { \sqrt { 1 + x ^ { 2 } } } \mathrm {~d} x\).
2. Explain how the trapezium rule might be used to give a better approximation to the integral given in part (a).

11 The region \(R\) enclosed by the lines \(x = 1 , x = 6 , y = 0\) and the curve $$y = \ln ( 8 - x )$$ is shown shaded in Figure 3 below. \begin{figure}[h] \end{figure} All distances are measured in centimetres.
11

1. Use a single trapezium to find an approximate value of the area of the shaded region, giving your answer in \(\mathrm { cm } ^ { 2 }\) to two decimal places.
   [0pt] [2 marks]
   \section*{Question 11 continues on the next page} 11
2. Shape \(B\) is made from four copies of region \(R\) as shown in Figure 4 below. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 4} \includegraphics[alt={},max width=\textwidth]{08e1f291-7052-40a5-b7b2-13fd1d0137c2-18_707_711_438_667}
   \end{figure} Shape \(B\) is cut from metal of thickness 2 mm
   The metal has a density of \(10.5 \mathrm {~g} / \mathrm { cm } ^ { 3 }\) Use the trapezium rule with six ordinates to calculate an approximate value of the mass of Shape B. Give your answer to the nearest gram.
   11
3. Without further calculation, give one reason why the mass found in part (b) may be:
   11 (c) (i) an underestimate.
   11 (c) (ii) an overestimate.

\includegraphics{figure_6} The diagram shows the curve with equation \(y = \sqrt{1 + 3\cos^2(\frac{1}{2}x)}\) for \(0 \leqslant x \leqslant \pi\). The region \(R\) is bounded by the curve, the axes and the line \(x = \pi\).

1. Use the trapezium rule with two intervals to find an approximation to the area of \(R\), giving your answer correct to 3 significant figures. [3]
2. The region \(R\) is rotated completely about the \(x\)-axis. Without using a calculator, find the exact volume of the solid produced. [5]

\includegraphics{figure_8} The diagram shows the curve \(y = 1.25^x\).

1. A point on the curve has y-coordinate 2. Calculate its x-coordinate. [3]
2. Use the trapezium rule with 4 intervals to estimate the area of the shaded region, bounded by the curve, the axes, and the line \(x = 4\). [4]
3. State, with a reason, whether the estimate found in part (ii) is an overestimate or an underestimate. [2]
4. Explain briefly how the trapezium rule could be used to find a more accurate estimate of the area of the shaded region. [1]

Oskar is designing a building. Fig. 12 shows his design for the end wall and the curve of the roof. The units for \(x\) and \(y\) are metres. \includegraphics{figure_12}

1. Use the trapezium rule with 5 strips to estimate the area of the end wall of the building. [4]
2. Oskar now uses the equation \(y = -0.001x^3 - 0.025x^2 + 0.6x + 9\), for \(0 \leq x \leq 15\), to model the curve of the roof.
   1. Calculate the difference between the height of the roof when \(x = 12\) given by this model and the data shown in Fig. 12. [2]
   2. Use integration to find the area of the end wall given by this model. [4]

\includegraphics{figure_2} The diagram shows the curve with equation \(y = 2^x\). Use the trapezium rule with four intervals, each of width 1, to estimate the area of the shaded region bounded by the curve, the \(x\)-axis and the lines \(x = -2\) and \(x = 2\). [4]

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]

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]

A particle of mass 4 kg moves horizontally in a straight line. At time \(t\) seconds the velocity of the particle is \(v\) m s\(^{-1}\) The following horizontal forces act on the particle: • a constant driving force of magnitude 1.8 newtons • another driving force of magnitude \(30\sqrt{t}\) newtons • a resistive force of magnitude \(0.08v^2\) newtons When \(t = 70\), \(v = 54\) Use Euler's method with a step length of 0.5 to estimate the velocity of the particle after 71 seconds. Give your answer to four significant figures. [6 marks]

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]