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A-LEVEL ALGEBRA · EDEXCEL · OCR
Area bound by a curve and the x-axis Exam Questions
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1CIE P1 November 2013 Q10
The diagram shows the curve y = (3 - 2x)^3 and the tangent to the curve at the point (1/2, 8). (i) Find the equation of this tangent, giving your answer in the form y = mx + c. [5] (ii) Find the area of the shaded region. [6]
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2OCR C2 June 2013 Q7
The diagram shows the curve y = x^(3/2) - 1, which crosses the x-axis at (1, 0), and the tangent to the curve at the point (4, 7). (i) Show that the integral from 1 to 4 of (x^(3/2) - 1) dx = 9 2/5. [4] (ii) Hence find the exact area of the shaded region enclosed by the curve, the tangent and the x-axis. [5]
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Part (b):
3EDEXCEL C2 January 2013 Q9
The finite region R, as shown in Figure 2, is bounded by the x-axis and the curve with equation y = 27 - 2x - 9 sqrt(x) - 16/x^2, x > 0. The curve crosses the x-axis at the points (1, 0) and (4, 0). (a) Complete the table below, by giving your values of y to 3 decimal places: x = 1, 1.5, 2, 2.5, 3, 3.5, 4 with y = 0, 5.866, blank, 5.210, blank, 1.856, 0. (2) (b) Use the trapezium rule with all the values in the completed table to find an approximate value for the area of R, giving your answer to 2 decimal places. (4) (c) Use integration to find the exact value for the area of R. (6)
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4EDEXCEL C2 June 2012 Q5
Figure 2 shows the line with equation y = 10 - x and the curve with equation y = 10x - x^2 - 8. The line and the curve intersect at the points A and B, and O is the origin. (a) Calculate the coordinates of A and the coordinates of B. (5) The shaded area R is bounded by the line and the curve, as shown in Figure 2. (b) Calculate the exact area of R. (7)
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5EDEXCEL C2 June 2011 Q9
The straight line with equation y = x + 4 cuts the curve with equation y = -x^2 + 2x + 24 at the points A and B, as shown in Figure 3. (a) Use algebra to find the coordinates of the points A and B. (4) The finite region R is bounded by the straight line and the curve and is shown shaded in Figure 3. (b) Use calculus to find the exact area of R. (7)
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Part (b):
6EDEXCEL C2 January 2011 Q4
Figure 1 shows a sketch of part of the curve C with equation y = (x + 1)(x - 5). The curve crosses the x-axis at the points A and B. (a) Write down the x-coordinates of A and B. (1) The finite region R, shown shaded in Figure 1, is bounded by C and the x-axis. (b) Use integration to find the area of R. (6)
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7EDEXCEL C2 January 2010 Q7
The curve C has equation y = x^2 - 5x + 4. It cuts the x-axis at the points L and M as shown in Figure 2. (a) Find the coordinates of the point L and the point M. (2) (b) Show that the point N (5, 4) lies on C. (1) (c) Find the integral of (x^2 - 5x + 4) dx. (2) The finite region R is bounded by LN, LM and the curve C as shown in Figure 2. (d) Use your answer to part (c) to find the exact value of the area of R. (5)
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Part (d):
8EDEXCEL C2 January 2009 Q2
Question image is broken on examsolutions.net (dead upload, reported). The video solution below walks through the question.
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9EDEXCEL C2 June 2008 Q8
Figure 2 shows a sketch of part of the curve with equation y = 10 + 8x + x^2 - x^3. The curve has a maximum turning point A. (a) Using calculus, show that the x-coordinate of A is 2. (3) The region R, shown shaded in Figure 2, is bounded by the curve, the y-axis and the line from O to A, where O is the origin. (b) Using calculus, find the exact area of R. (8)
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10EDEXCEL C2 January 2008 Q7
In Figure 2 the curve C has equation y = 6x - x^2 and the line L has equation y = 2x. (a) Show that the curve C intersects the x-axis at x = 0 and x = 6. (1) (b) Show that the line L intersects the curve C at the points (0, 0) and (4, 8). (3) The region R, bounded by the curve C and the line L, is shown shaded in Figure 2. (c) Use calculus to find the area of R. (6)
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Part (b):
Part (c):
11EDEXCEL C2 January 2007 Q7
Figure 1 shows a sketch of part of the curve C with equation y = x(x - 1)(x - 5). Use calculus to find the total area of the finite region, shown shaded in Figure 1, that is between x = 0 and x = 2 and is bounded by C, the x-axis and the line x = 2. (9)
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12Edexcel A Level June 2022 Paper 1 Q16
Figure 6 shows a sketch of the curve C with parametric equations x = 8 sin^2 t, y = 2 sin 2t + 3 sin t, 0 <= t <= pi/2. The region R, shown shaded in Figure 6, is bounded by C, the x-axis and the line with equation x = 4. (a) Show that the area of R is given by the integral from 0 to a of (8 - 8 cos 4t + 48 sin^2 t cos t) dt, where a is a constant to be found. (5) (b) Hence, using algebraic integration, find the exact area of R. (4)
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13Edexcel AS June 2023 Paper 1 Q5
In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. The finite region R, shown shaded in Figure 2, is bounded by the curve with equation y = 4x^2 + 3, the y-axis and the line with equation y = 23. Show that the exact area of R is k sqrt(5), where k is a rational constant to be found. (5)
14Edexcel A Level June 2024 Paper 1 Q10
In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. Figure 3 shows a sketch of part of the curve with equation y = 8x - x^(5/2), x >= 0. The curve crosses the x-axis at the point A. (a) Verify that the x coordinate of A is 4. (1) The line l1 is the tangent to the curve at A. (b) Use calculus to show that an equation of line l1 is 12x + y = 48. (3) The line l2 has equation y = 8x. The region R, shown shaded in Figure 3, is bounded by the curve, the line l1 and the line l2. (c) Use algebraic integration to find the exact area of R. (5)
