1.08d Evaluate definite integrals: between limits

575 questions

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Edexcel C2 Q2
10 marks Moderate -0.8
  1. Differentiate with respect to x $$2x^3 + \sqrt{x} + \frac{x^2 + 2x}{x^2}.$$ [5 marks]
  2. Evaluate $$\int_1^4 \left(\frac{x}{2} + \frac{1}{x^2}\right) dx.$$ [5 marks]
Edexcel C2 Q5
10 marks Moderate -0.8
  1. Differentiate \(2x^2 + \sqrt{x} + \frac{x^2 + 2x}{x^2}\) with respect to \(x\) [5]
  2. Evaluate \(\int_1^4 \left(\frac{x}{2} + \frac{1}{x^2}\right) dx\). [5]
OCR C2 Q3
7 marks Moderate -0.8
  1. Find \(\int (2x + 1)(x + 3) \, dx\). [4]
  2. Evaluate \(\int_0^9 \frac{1}{\sqrt{x}} \, dx\). [3]
OCR C2 2007 January Q10
10 marks Standard +0.3
\includegraphics{figure_10} The diagram shows the graph of \(y = 1 - 3x^{-\frac{1}{2}}\).
  1. Verify that the curve intersects the \(x\)-axis at \((9, 0)\). [1]
  2. The shaded region is enclosed by the curve, the \(x\)-axis and the line \(x = a\) (where \(a > 9\)). Given that the area of the shaded region is 4 square units, find the value of \(a\). [9]
OCR MEI C2 2010 January Q11
12 marks Moderate -0.3
Fig. 11 shows the cross-section of a school hall, with measurements of the height in metres taken at 1.5 m intervals from O. \includegraphics{figure_11}
  1. Use the trapezium rule with 8 strips to calculate an estimate of the area of the cross-section. [4]
  2. Use 8 rectangles to calculate a lower bound for the area of the cross-section. [2]
The curve of the roof may be modelled by \(y = -0.013x^3 + 0.16x^2 - 0.082x + 2.4\), where \(x\) metres is the horizontal distance from O across the hall, and \(y\) metres is the height.
  1. Use integration to find the area of the cross-section according to this model. [4]
  2. Comment on the accuracy of this model for the height of the hall when \(x = 7.5\). [2]
OCR MEI C2 2006 June Q4
5 marks Moderate -0.8
Find \(\int_1^2 \left( x^4 - \frac{3}{x^2} + 1 \right) dx\), showing your working. [5]
OCR MEI C2 2008 June Q12
12 marks Moderate -0.8
\includegraphics{figure_12} A water trough is a prism 2.5 m long. Fig. 12 shows the cross-section of the trough, with the depths in metres at 0.1 m intervals across the trough. The trough is full of water.
  1. Use the trapezium rule with 5 strips to calculate an estimate of the area of cross-section of the trough. Hence estimate the volume of water in the trough. [5]
  2. A computer program models the curve of the base of the trough, with axes as shown and units in metres, using the equation \(y = 8x^3 - 3x^2 - 0.5x - 0.15\), for \(0 \leq x \leq 0.5\). Calculate \(\int_0^{0.5} (8x^3 - 3x^2 - 0.5x - 0.15) \, \text{d}x\) and state what this represents. Hence find the volume of water in the trough as given by this model. [7]
OCR MEI C2 2010 June Q5
4 marks Moderate -0.8
Find \(\int_{2}^{5} \left(1 - \frac{6}{x^3}\right) dx\). [4]
OCR MEI C2 2014 June Q12
10 marks Moderate -0.3
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]
OCR MEI C2 2016 June Q9
11 marks Standard +0.3
Fig. 9 shows the cross-section of a straight, horizontal tunnel. The \(x\)-axis from 0 to 6 represents the floor of the tunnel. \includegraphics{figure_9} With axes as shown, and units in metres, the roof of the tunnel passes through the points shown in the table.
\(x\)0123456
\(y\)04.04.95.04.94.00
The length of the tunnel is 50 m.
  1. Use the trapezium rule with 6 strips to estimate the area of cross-section of the tunnel. Hence estimate the volume of earth removed in digging the tunnel. [4]
  2. An engineer models the height of the roof of the tunnel using the curve \(y = \frac{x}{81}(108x - 54x^2 + 12x^3 - x^4)\). This curve is symmetrical about \(x = 3\).
    1. Show that, according to this model, a vehicle of rectangular cross-section which is 3.6 m wide and 4.4 m high would not be able to pass through the tunnel. [2]
    2. Use integration to calculate the area of the cross-section given by this model. Hence obtain another estimate of the volume of earth removed in digging the tunnel. [5]
OCR C2 Q7
9 marks Moderate -0.8
  1. Find $$\int \left( x + 5 + \frac{3}{\sqrt{x}} \right) dx.$$ [4]
  2. Evaluate $$\int_{-2}^{0} (3x - 1)^2 dx.$$ [5]
OCR C2 Q6
8 marks Moderate -0.3
\includegraphics{figure_6} The diagram shows the curve with equation \(y = 4x^{\frac{1}{3}} - x\), \(x \geq 0\). The curve meets the \(x\)-axis at the origin and at the point \(A\) with coordinates \((a, 0)\).
  1. Show that \(a = 8\). [3]
  2. Find the area of the finite region bounded by the curve and the positive \(x\)-axis. [5]
OCR C2 Q9
13 marks Moderate -0.3
  1. Evaluate $$\int_1^3 (3 - \sqrt{x})^2 \, dx,$$ giving your answer in the form \(a + b\sqrt{3}\), where \(a\) and \(b\) are integers. [6]
  2. The gradient of a curve is given by $$\frac{dy}{dx} = 3x^2 + 4x + k,$$ where \(k\) is a constant. Given that the curve passes through the points \((0, -2)\) and \((2, 18)\), show that \(k = 2\) and find an equation for the curve. [7]
OCR C2 Q8
12 marks Moderate -0.3
  1. The gradient of a curve is given by $$\frac{dy}{dx} = 3 - \frac{2}{x^2}, \quad x \neq 0.$$ Find an equation for the curve given that it passes through the point \((2, 6)\). [6]
  2. Show that $$\int_2^3 (6\sqrt{x} - \frac{4}{\sqrt{x}}) \, dx = k\sqrt{3},$$ where \(k\) is an integer to be found. [6]
OCR MEI C2 Q4
3 marks Moderate -0.8
Find \(\int_2^5 (2x^3 + 3) dx\). [3]
OCR MEI C2 Q6
4 marks Moderate -0.3
Find \(\int_2^5 \left(1 - \frac{6}{x^3}\right) dx\). [4]
OCR MEI C2 Q7
4 marks Easy -1.2
Find \(\int_1^2 (12x^5 + 5) dx\). [4]
OCR MEI C2 Q10
5 marks Moderate -0.8
Find \(\int_1^2 \left(x^4 - \frac{3}{x^2} + 1\right) dx\), showing your working. [5]
OCR MEI C2 Q2
11 marks Moderate -0.3
Fig. 11 shows the curve \(y = x^3 - 3x^2 - x + 3\). \includegraphics{figure_11}
  1. Use calculus to find \(\int_{-1}^{3} (x^3 - 3x^2 - x + 3) dx\) and state what this represents. [6]
  2. Find the \(x\)-coordinates of the turning points of the curve \(y = x^3 - 3x^2 - x + 3\), giving your answers in surd form. Hence state the set of values of \(x\) for which \(y = x^3 - 3x^2 - x + 3\) is a decreasing function. [5]
AQA C3 2011 June Q9
11 marks Standard +0.3
  1. Use integration by parts to find \(\int x\ln x \, dx\). [3]
  2. Given that \(y = (\ln x)^2\), find \(\frac{dy}{dx}\). [2]
  3. The diagram shows part of the curve with equation \(y = \sqrt{x\ln x}\). \includegraphics{figure_9} The shaded region \(R\) is bounded by the curve \(y = \sqrt{x\ln x}\), the line \(x = e\) and the \(x\)-axis from \(x = 1\) to \(x = e\). Find the volume of the solid generated when the region \(R\) is rotated through 360° about the \(x\)-axis, giving your answer in an exact form. [6]
OCR C3 Q4
8 marks Standard +0.2
  1. \includegraphics{figure_4a} The diagram shows the curve \(y = \frac{2}{\sqrt{x}}\). The region \(R\), shaded in the diagram, is bounded by the curve and by the lines \(x = 1\), \(x = 5\) and \(y = 0\). The region \(R\) is rotated completely about the \(x\)-axis. Find the exact volume of the solid formed. [4]
  2. Use Simpson's rule, with 4 strips, to find an approximate value for $$\int_1^5 \sqrt{(x^2 + 1)} \, dx,$$ giving your answer correct to 3 decimal places. [4]
OCR C3 Q8
16 marks Standard +0.3
\includegraphics{figure_8} The diagram shows part of each of the curves \(y = e^{\frac{1}{3}x}\) and \(y = \sqrt[3]{(3x + 8)}\). The curves meet, as shown in the diagram, at the point \(P\). The region \(R\), shaded in the diagram, is bounded by the two curves and by the \(y\)-axis.
  1. Show by calculation that the \(x\)-coordinate of \(P\) lies between 5.2 and 5.3. [3]
  2. Show that the \(x\)-coordinate of \(P\) satisfies the equation \(x = \frac{2}{3} \ln(3x + 8)\). [2]
  3. Use an iterative formula, based on the equation in part (ii), to find the \(x\)-coordinate of \(P\) correct to 2 decimal places. [3]
  4. Use integration, and your answer to part (iii), to find an approximate value of the area of the region \(R\). [5]
OCR C3 Q1
4 marks Moderate -0.5
Show that \(\int_2^8 \frac{3}{x} \, dx = \ln 64\). [4]
OCR C3 Q5
8 marks Standard +0.8
\includegraphics{figure_5} The diagram shows the curves \(y = (1 - 2x)^5\) and \(y = e^{2x-1} - 1\). The curves meet at the point \((\frac{1}{2}, 0)\). Find the exact area of the region (shaded in the diagram) bounded by the \(y\)-axis and by part of each curve. [8]
OCR C3 Q7
10 marks Moderate -0.3
  1. Find the exact value of \(\int_1^2 \frac{2}{(4x - 1)^2} \, dx\). [4]
  2. \includegraphics{figure_7b} The diagram shows part of the curve \(y = \frac{1}{x}\). The point \(P\) has coordinates \((a, \frac{1}{a})\) and the point \(Q\) has coordinates \((2a, \frac{1}{2a})\), where \(a\) is a positive constant. The point \(R\) is such that \(PR\) is parallel to the \(x\)-axis and \(QR\) is parallel to the \(y\)-axis. The region shaded in the diagram is bounded by the curve and by the lines \(PR\) and \(QR\). Show that the area of this shaded region is \(\ln(\frac{4}{e})\). [6]