1.09f Trapezium rule: numerical integration

8 Fig. 8.1 shows the cross-section of a straight driveway 4 m wide made from tarmac. \begin{figure}[h] \end{figure} The height \(h \mathrm {~m}\) of the cross-section at a displacement \(x \mathrm {~m}\) from the middle is modelled by \(\mathrm { h } = \frac { 0.2 } { 1 + \mathrm { x } ^ { 2 } }\) for \(- 2 \leqslant x \leqslant 2\). A lower bound of \(0.3615 \mathrm {~m} ^ { 2 }\) is found for the area of the cross-section using rectangles as shown in Fig. 8.2. \begin{figure}[h] \end{figure}

1. Use a similar method to find an upper bound for the area of the cross-section.
2. Use the trapezium rule with 4 strips to estimate \(\int _ { 0 } ^ { 2 } \frac { 0.2 } { 1 + x ^ { 2 } } d x\).
3. The driveway is 10 m long. Use your answer in part (b) to find an estimate of the volume of tarmac needed to make the driveway.

6 Fig. 6 shows a partially completed spreadsheet.
This spreadsheet uses the trapezium rule with four strips to estimate \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \sqrt { 1 + \sin x } \mathrm {~d} x\). \begin{table}[h] \end{table}

1. Show how the value in cell B3 is calculated.
2. Show how the values in cells D2 to D6 are used to calculate the value in cell E7.
3. Complete the calculation to estimate \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \sqrt { 1 + \sin x } \mathrm {~d} x\). Give your answer to 3 significant figures.

4 Fig. 4 shows the graph of \(y = \sqrt { 1 + x ^ { 3 } }\). \begin{figure}[h] \end{figure}

1. Use the trapezium rule with \(h = 0.5\) to find an estimate of \(\int _ { - 1 } ^ { 0 } \sqrt { 1 + x ^ { 3 } } \mathrm {~d} x\), giving your answer correct to 6 decimal places.
2. State whether your answer to part (a) is an under-estimate or an over-estimate, justifying your answer.

8 The diagram shows a sketch of the curve with equation \(y = 3 ^ { x } + 1\). \includegraphics[max width=\textwidth, alt={}, center]{4a4d4dcd-4137-427d-834f-ac2fe83f8aeb-5_535_1011_411_513} The curve intersects the \(y\)-axis at the point \(A\).

1. Write down the \(y\)-coordinate of point \(A\).
2. 1. Use the trapezium rule with five ordinates (four strips) to find an approximation for \(\int _ { 0 } ^ { 1 } \left( 3 ^ { x } + 1 \right) \mathrm { d } x\), giving your answer to three significant figures.
      (4 marks)
   2. By considering the graph of \(y = 3 ^ { x } + 1\), explain with the aid of a diagram whether your approximation will be an overestimate or an underestimate of the true value of \(\int _ { 0 } ^ { 1 } \left( 3 ^ { x } + 1 \right) \mathrm { d } x\).
      (2 marks)
3. The line \(y = 5\) intersects the curve \(y = 3 ^ { x } + 1\) at the point \(P\). By solving a suitable equation, find the \(x\)-coordinate of the point \(P\). Give your answer to four decimal places.
   (4 marks)
4. The curve \(y = 3 ^ { x } + 1\) is reflected in the \(y\)-axis to give the curve with equation \(y = \mathrm { f } ( x )\). Write down an expression for \(\mathrm { f } ( x )\).
   (1 mark)

2

1. Use the trapezium rule with five ordinates (four strips) to find an approximate value for $$\int _ { 0 } ^ { 4 } \frac { 1 } { x ^ { 2 } + 1 } \mathrm {~d} x$$ giving your answer to four significant figures.
2. State how you could obtain a better approximation to the value of the integral using the trapezium rule.

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

2

1. Use the trapezium rule with four ordinates (three strips) to find an approximate value for $$\int _ { 1.5 } ^ { 6 } x ^ { 2 } \sqrt { x ^ { 2 } - 1 } \mathrm {~d} x$$ giving your answer to three significant figures.
2. State how you could obtain a better approximation to the value of the integral using the trapezium rule.
   (1 mark)

6

1. Sketch the curve with equation \(y = 2 ^ { x }\), indicating the coordinates of any point where the curve intersects the coordinate axes.
2. 1. Use the trapezium rule with five ordinates (four strips) to find an approximate value for \(\int _ { 0 } ^ { 2 } 2 ^ { x } \mathrm {~d} x\), giving your answer to three significant figures.
   2. State how you could obtain a better approximation to the value of the integral using the trapezium rule.
3. Describe a geometrical transformation that maps the graph of \(y = 2 ^ { x }\) onto the graph of \(y = 2 ^ { x + 7 } + 3\).
4. The curve \(y = 2 ^ { x + k } + 3\) intersects the \(y\)-axis at the point \(A ( 0,8 )\). Show that \(k = \log _ { m } n\), where \(m\) and \(n\) are integers.

4

1. Use the trapezium rule with four ordinates (three strips) to find an approximate value for \(\int _ { 0 } ^ { 1.5 } \sqrt { 27 x ^ { 3 } + 4 } \mathrm {~d} x\), giving your answer to three significant figures.
2. The curve with equation \(y = \sqrt { 27 x ^ { 3 } + 4 }\) is stretched parallel to the \(x\)-axis with scale factor 3 to give the curve with equation \(y = \mathrm { g } ( x )\). Write down an expression for \(\mathrm { g } ( x )\).
   (2 marks)
   \includegraphics[max width=\textwidth, alt={}]{1c06ba04-575c-4eb8-b4aa-0a7510838cd2-05_1988_1717_719_150}

2

1. Use the trapezium rule with five ordinates (four strips) to find an approximate value for $$\int _ { 0 } ^ { 4 } \frac { 2 ^ { x } } { x + 1 } \mathrm {~d} x$$ giving your answer to three significant figures.
2. State how you could obtain a better approximation to the value of the integral using the trapezium rule.

2

1. Use the trapezium rule with five ordinates (four strips) to find an approximate value for $$\int _ { 1 } ^ { 5 } \frac { 1 } { x ^ { 2 } + 1 } \mathrm {~d} x$$ giving your answer to three significant figures.
2. 1. Find \(\int \left( x ^ { - \frac { 3 } { 2 } } + 6 x ^ { \frac { 1 } { 2 } } \right) \mathrm { d } x\), giving the coefficient of each term in its simplest form.
   2. Hence find the value of \(\int _ { 1 } ^ { 4 } \left( x ^ { - \frac { 3 } { 2 } } + 6 x ^ { \frac { 1 } { 2 } } \right) \mathrm { d } x\).

4 The diagram shows a curve \(C\) with equation \(y = \sqrt { x }\). The point \(O\) is the origin \(( 0,0 )\). \includegraphics[max width=\textwidth, alt={}, center]{37627fc4-a90b-4f3b-9b10-0a9e200f8485-3_488_1136_1009_443} The region bounded by the curve \(C\), the \(x\)-axis and the vertical lines \(x = 1\) and \(x = 4\) is shown shaded in the diagram.

1. 1. Write \(\sqrt { x }\) in the form \(x ^ { p }\), where \(p\) is a constant.
   2. Find \(\int \sqrt { x } \mathrm {~d} x\).
   3. Hence find the area of the shaded region.
2. The point on \(C\) for which \(x = 4\) is \(P\). The tangent to \(C\) at the point \(P\) intersects the \(x\)-axis and the \(y\)-axis at the points \(A\) and \(B\) respectively.
   1. Find an equation for the tangent to the curve \(C\) at the point \(P\).
   2. Find the area of the triangle \(A O B\).
3. Describe the single geometrical transformation by which the curve with equation \(y = \sqrt { x - 1 }\) can be obtained from the curve \(C\).
4. Use the trapezium rule with four ordinates (three strips) to find an approximation for \(\int _ { 1 } ^ { 4 } \sqrt { x - 1 } \mathrm {~d} x\), giving your answer to three significant figures.

6 The diagram shows a sketch of the curve with equation \(y = 27 - 3 ^ { x }\). \includegraphics[max width=\textwidth, alt={}, center]{f066f68a-e739-4da3-8ec1-e221461146b0-4_933_1074_376_484} The curve \(y = 27 - 3 ^ { x }\) intersects the \(y\)-axis at the point \(A\) and the \(x\)-axis at the point \(B\).

1. 1. Find the \(y\)-coordinate of point \(A\).
   2. Verify that the \(x\)-coordinate of point \(B\) is 3 .
2. The region, \(R\), bounded by the curve \(y = 27 - 3 ^ { x }\) and the coordinate axes is shaded. Use the trapezium rule with four ordinates (three strips) to find an approximate value for the area of \(R\).
3. 1. Use logarithms to solve the equation \(3 ^ { x } = 13\), giving your answer to four decimal places.
   2. The line \(y = k\) intersects the curve \(y = 27 - 3 ^ { x }\) at the point where \(3 ^ { x } = 13\). Find the value of \(k\).
4. 1. Describe the single geometrical transformation by which the curve with equation \(y = - 3 ^ { x }\) can be obtained from the curve \(y = 27 - 3 ^ { x }\).
   2. Sketch the curve \(y = - 3 ^ { x }\).

8 The diagram shows a sketch of the curve with equation \(y = 6 ^ { x }\). \includegraphics[max width=\textwidth, alt={}, center]{a2525df8-dbd0-4b69-b6bb-f8ef6f96f7dc-5_403_506_370_769}

1. 1. Use the trapezium rule with five ordinates (four strips) to find an approximate value for \(\int _ { 0 } ^ { 2 } 6 ^ { x } \mathrm {~d} x\), giving your answer to three significant figures.
   2. Explain, with the aid of a diagram, whether your approximate value will be an overestimate or an underestimate of the true value of \(\int _ { 0 } ^ { 2 } 6 ^ { x } \mathrm {~d} x\).
2. 1. Describe a single geometrical transformation that maps the graph of \(y = 6 ^ { x }\) onto the graph of \(y = 6 ^ { 3 x }\).
   2. The line \(y = 84\) intersects the curve \(y = 6 ^ { 3 x }\) at the point \(A\). By using logarithms, find the \(x\)-coordinate of \(A\), giving your answer to three decimal places.
      (4 marks)
3. The graph of \(y = 6 ^ { x }\) is translated by \(\left[ \begin{array} { c } 1 \\ - 2 \end{array} \right]\) to give the graph of the curve with equation \(y = \mathrm { f } ( x )\). Write down an expression for \(\mathrm { f } ( x )\).

8 The diagram shows a sketch of the curve \(y = 2 ^ { 4 x }\). \includegraphics[max width=\textwidth, alt={}, center]{f9a7a4dd-f7fd-4135-8872-2c1270d46a14-9_435_814_374_623} The curve intersects the \(y\)-axis at the point \(A\).

1. Find the value of the \(y\)-coordinate of \(A\).
2. Use the trapezium rule with six ordinates (five strips) to find an approximate value for \(\int _ { 0 } ^ { 1 } 2 ^ { 4 x } \mathrm {~d} x\), giving your answer to two decimal places.
3. Describe the geometrical transformation that maps the graph of \(y = 2 ^ { 4 x }\) onto the graph of \(y = 2 ^ { 4 x - 3 }\).
4. The curve \(y = 2 ^ { 4 x }\) is translated by the vector \(\left[ \begin{array} { c } 1 \\ - \frac { 1 } { 2 } \end{array} \right]\) to give the curve \(y = \mathrm { g } ( x )\). The curve \(y = \mathrm { g } ( x )\) crosses the \(x\)-axis at the point \(Q\). Find the \(x\)-coordinate of \(Q\).
5. 1. Given that $$\log _ { a } k = 3 \log _ { a } 2 + \log _ { a } 5 - \log _ { a } 4$$ show that \(k = 10\).
   2. The line \(y = \frac { 5 } { 4 }\) crosses the curve \(y = 2 ^ { 4 x - 3 }\) at the point \(P\). Show that the \(x\)-coordinate of \(P\) is \(\frac { 1 } { 4 \log _ { 10 } 2 }\).

9 The diagram shows part of a curve whose equation is \(y = \log _ { 10 } \left( x ^ { 2 } + 1 \right)\). \includegraphics[max width=\textwidth, alt={}, center]{a5fa3066-e330-46d0-98e3-92d438ed6f61-5_355_451_367_799}

1. Use the trapezium rule with five ordinates (four strips) to find an approximate value for $$\int _ { 0 } ^ { 1 } \log _ { 10 } \left( x ^ { 2 } + 1 \right) d x$$ giving your answer to three significant figures.
2. The graph of \(y = 2 \log _ { 10 } x\) can be transformed into the graph of \(y = 1 + 2 \log _ { 10 } x\) by means of a translation. Write down the vector of the translation.
3. 1. Show that \(\log _ { 10 } \left( 10 x ^ { 2 } \right) = 1 + 2 \log _ { 10 } x\).
   2. Show that the graph of \(y = 2 \log _ { 10 } x\) can also be transformed into the graph of \(y = 1 + 2 \log _ { 10 } x\) by means of a stretch, and describe the stretch.
   3. The curve with equation \(y = 1 + 2 \log _ { 10 } x\) intersects the curve \(y = \log _ { 10 } \left( x ^ { 2 } + 1 \right)\) at the point \(P\). Given that the \(x\)-coordinate of \(P\) is positive, find the gradient of the line \(O P\), where \(O\) is the origin. Give your answer in the form \(\log _ { 10 } \left( \frac { a } { b } \right)\), where \(a\) and \(b\) are integers.

5

1. Use the trapezium rule with five ordinates (four strips) to find an approximate value for \(\int _ { 0 } ^ { 2 } \sqrt { 8 x ^ { 3 } + 1 } \mathrm {~d} x\), giving your answer to three significant figures.
2. Describe the single transformation that maps the graph of \(y = \sqrt { 8 x ^ { 3 } + 1 }\) onto the graph of \(y = \sqrt { x ^ { 3 } + 1 }\).
3. The curve with equation \(y = \sqrt { x ^ { 3 } + 1 }\) is translated by \(\left[ \begin{array} { c } 2 \\ - 0.7 \end{array} \right]\) to give the curve with equation \(y = \mathrm { g } ( x )\). Find the value of \(\mathrm { g } ( 4 )\).
   (3 marks)

9 A curve has equation \(y = 3 \times 12 ^ { x }\).

1. The point ( \(k , 6\) ) lies on the curve \(y = 3 \times 12 ^ { x }\). Use logarithms to find the value of \(k\), giving your answer to three significant figures.
2. Use the trapezium rule with four ordinates (three strips) to find an approximate value for \(\int _ { 0 } ^ { 1.5 } 3 \times 12 ^ { x } \mathrm {~d} x\), giving your answer to two significant figures.
3. The curve \(y = 3 \times 12 ^ { x }\) is translated by the vector \(\left[ \begin{array} { l } 1 \\ p \end{array} \right]\) to give the curve \(y = \mathrm { f } ( x )\). Given that the curve \(y = \mathrm { f } ( x )\) passes through the origin ( 0,0 ), find the value of the constant \(p\).
4. The curve with equation \(y = 2 ^ { 2 - x }\) intersects the curve \(y = 3 \times 12 ^ { x }\) at the point \(T\). Show that the \(x\)-coordinate of \(T\) can be written in the form \(\frac { 2 - \log _ { 2 } 3 } { q + \log _ { 2 } 3 }\), where \(q\) is an integer. State the value of \(q\).
   [0pt] [5 marks]
   \includegraphics[max width=\textwidth, alt={}]{30ccdbe9-0c91-4011-a3f9-3ce01862215d-20_2288_1707_221_153}

7 The diagram shows a sketch of two curves. \includegraphics[max width=\textwidth, alt={}, center]{24641e66-b73b-4323-98c8-349727151aba-14_448_527_370_762} The equations of the two curves are \(y = 1 + \sqrt { x }\) and \(y = 4 ^ { \frac { x } { 9 } }\).
The curves meet at the points \(P ( 0,1 )\) and \(Q ( 9,4 )\).

1. 1. Describe the geometrical transformation that maps the graph of \(y = \sqrt { x }\) onto the graph of \(y = 1 + \sqrt { x }\).
   2. Describe the geometrical transformation that maps the graph of \(y = 4 ^ { x }\) onto the graph of \(y = 4 ^ { \frac { x } { 9 } }\).
2. 1. Given that \(\int _ { 0 } ^ { 9 } \sqrt { x } \mathrm {~d} x = 18\), find the value of \(\int _ { 0 } ^ { 9 } ( 1 + \sqrt { x } ) \mathrm { d } x\).
   2. Use the trapezium rule with five ordinates (four strips) to find an approximate value for \(\int _ { 0 } ^ { 9 } 4 ^ { \frac { x } { 9 } } \mathrm {~d} x\). Give your answer to one decimal place.
   3. Hence find an approximate value for the area of the shaded region bounded by the two curves and state, with an explanation, whether your approximation will be an overestimate or an underestimate of the true value for the area of the shaded region.
      [0pt] [3 marks]

5

1. Use the trapezium rule with four ordinates (three strips) to find an approximate value for \(\int _ { 2 } ^ { 11 } \sqrt { x ^ { 2 } + 9 } \mathrm {~d} x\). Give your answer to one decimal place.
2. Describe the geometrical transformation that maps the graph of \(y = \sqrt { x ^ { 2 } + 9 }\) onto the graph of :
   1. \(y = 5 + \sqrt { x ^ { 2 } + 9 }\);
   2. \(y = 3 \sqrt { x ^ { 2 } + 1 }\).

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.

6. \begin{figure}[h] \end{figure} Figure 1 shows the curve with equation \(y = \left( x - \log _ { 10 } x \right) ^ { 2 } , x > 0\).

1. Copy and complete the table below for points on the curve, giving the \(y\) values to 2 decimal places.

   \(x\)	2	3	4	5	6
   \(y\)	2.89	6.36

   The shaded area is bounded by the curve, the \(x\)-axis and the lines \(x = 2\) and \(x = 6\).
2. Use the trapezium rule with all the values in your table to estimate the area of the shaded region.
3. State, with a reason, whether your answer to part (b) is an under-estimate or an over-estimate of the true area.