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Integration I · CAPE Pure Mathematics Unit 1

133 past-paper questions on Integration I, part of Calculus I, from every CAPE Pure Mathematics Unit 1 paper on Quelpr.

Practise Integration I
  1. Q361 mark · multiple choice· CAPE Pure Mathematics Unit 1 · 2008 (Rest of Region) · Paper 1An expression for obtaining the volume generated by rotating the bounded, shaded region through 360^\circ about the x-axis is
  2. Q381 mark · multiple choice· CAPE Pure Mathematics Unit 1 · 2008 (Rest of Region) · Paper 1If \frac{dy}{dx} = \cos x, then
  3. Q391 mark · multiple choice· CAPE Pure Mathematics Unit 1 · 2008 (Rest of Region) · Paper 1The area of the finite region shaded in the diagram is
  4. Q421 mark · multiple choice· CAPE Pure Mathematics Unit 1 · 2008 (Rest of Region) · Paper 1\int_{0}^{\frac{\pi}{2}} 2\cos 5x \, dx is
  5. Q431 mark · multiple choice· CAPE Pure Mathematics Unit 1 · 2008 (Rest of Region) · Paper 1Given \frac{dy}{dx} = 2x, a sketch of y versus x may be represented by I. II. III. IV.
  6. Q411 mark · multiple choice· CAPE Pure Mathematics Unit 1 · 2008 (T&T) · Paper 1\int_0^{\frac{\pi}{2}} \cos 5x\,\mathrm{d}x is
  7. Q421 mark · multiple choice· CAPE Pure Mathematics Unit 1 · 2008 (T&T) · Paper 1If f'(x) = \sin x, then f(x) =
  8. Q431 mark · multiple choice· CAPE Pure Mathematics Unit 1 · 2008 (T&T) · Paper 1\int [2x^{2n-1} + x^{3n-1}]\,\mathrm{d}x is equal to
  9. Q441 mark · multiple choice· CAPE Pure Mathematics Unit 1 · 2008 (T&T) · Paper 1Given that \int_2^5 4f(x)\,\mathrm{d}x = 9, the value of \int_2^5 [3 - f(x)]\,\mathrm{d}x is
  10. Q451 mark · multiple choice· CAPE Pure Mathematics Unit 1 · 2008 (T&T) · Paper 1The TOTAL shaded area in the diagram below is given by
  11. 6(a)5 marks· CAPE Pure Mathematics Unit 1 · 2022 · Paper 2Using the substitution u = x^2 + 1, determine \int 4x(x^2 + 1)^5\, dx.
  12. 6(b)6 marks· CAPE Pure Mathematics Unit 1 · 2022 · Paper 2The diagram below shows two curves, y = x + x^2 and y = x(3 - x). Calculate the area of the shaded region.
  13. 6(c)6 marks· CAPE Pure Mathematics Unit 1 · 2022 · Paper 2Calculate the volume of the solid generated by revolving the region bounded by the line y = 6x and the parabola y = 6x^2 about the x-axis.
  14. 6(d)8 marks· CAPE Pure Mathematics Unit 1 · 2022 · Paper 2Solve the differential equation \frac{dy}{dx} = \frac{x + 4x^2}{y^2}, given that y = 6 when x = 0.
  15. 15(b)4 marks· CAPE Pure Mathematics Unit 1 · May/June 2005 · Paper 1Find the TOTAL area bounded by the curve shown above, the x-axis and the lines x = -1 and x = 2.
  16. 14(a)1 mark· CAPE Pure Mathematics Unit 1 · May/June 2006 · Paper 1Express the shaded area, A, as the difference of two definite integrals.
  17. 14(b)2 marks· CAPE Pure Mathematics Unit 1 · May/June 2006 · Paper 1Hence, show that A = 16 int_2^3 x^(-2) dx - 1/2 int_2^3 x dx + int_2^3 dx.
  18. 14(c)3 marks· CAPE Pure Mathematics Unit 1 · May/June 2006 · Paper 1Find the value of A.
  19. 15(a)2 marks· CAPE Pure Mathematics Unit 1 · May/June 2006 · Paper 1Show that int_0^pi x sin x dx = int_0^pi (pi - x) sin x dx.
  20. 15(b)(i)2 marks· CAPE Pure Mathematics Unit 1 · May/June 2006 · Paper 1Hence, show that int_0^pi x sin x dx = pi int_0^pi sin x dx - int_0^pi x sin x dx.
  21. 15(b)(ii)5 marks· CAPE Pure Mathematics Unit 1 · May/June 2006 · Paper 1Hence, show that int_0^pi x sin x dx = pi.
  22. Q341 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2009 · Paper 1Given that \int_2^4 4f(x)\,dx = 9, the value of \int_4^2 3f(x)\,dx is
  23. Q351 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2009 · Paper 1The portion of the curve y = 2x between x = a and x = b is rotated 360^\circ (2\pi radians) about the x-axis. The volume, v, of the solid generated is BEST represented as
  24. Q411 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2009 · Paper 1\int_0^\frac{\pi}{4} \sec^2 x\,dx =
  25. Q421 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2009 · Paper 1The area of R is
  26. Q431 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2009 · Paper 1If the rate of change of y with respect to x is 3x^2 + \frac{4}{x^3} then y is equal to
  27. Q341 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2010 · Paper 1If \int_2^4 f(x)\,\mathrm{d}x = 12, what is the value of \int_2^3 f(x)\,\mathrm{d}x + \int_1^2 (f(x) - 1)\,\mathrm{d}x?
  28. Q351 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2010 · Paper 1Item 35 refers to the diagram below. The finite region R is enclosed by the curve y = x^2, the y-axis and the line y = 3 as shown in the diagram above. This region is rotated completely about the y-axis to…
  29. Q411 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2010 · Paper 1If f(x) = \begin{cases} 2 & \text{for } x \le 2 \\ -1 & \text{for } x > 2 \end{cases}, then \int_1^4 f(x)\,\mathrm{d}x equals
  30. Q431 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2010 · Paper 1At time t years, the growth of a certain country's gross national product, G, is given by the equation \frac{\mathrm{d}G}{\mathrm{d}t} = 5 + \cos t. At the beginning of the year 1990, the gross national product…
  31. Q451 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2010 · Paper 1Water is leaking from a tank. The rate of change in volume of water in the tank with respect to time, t, is inversely proportional to the volume, V, of water in the tank. If k is a positive constant of…
  32. Q351 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2011 · Paper 1The volume generated by rotating the bounded, shaded region through 360^\circ about the x-axis is
  33. Q381 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2011 · Paper 1If \frac{dy}{dx} = \cos x then
  34. Q411 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2011 · Paper 1Given that \int_2^5 4 f(x)\,dx = 9, the value of \int_2^5 3 f(x)\,dx is
  35. Q341 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2012 · Paper 1Given that \int_3^5 4f(x)\,dx = 9, the value of \int_3^5 f(x)\,dx is
  36. Q351 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2012 · Paper 1An expression for obtaining the volume generated by rotating the bounded, shaded region through 360^\circ about the x-axis is
  37. Q401 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2012 · Paper 1\int_0^{\frac{\pi}{4}} \sec^2 x\,dx =
  38. Q421 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2012 · Paper 1\int_0^{\frac{\pi}{2}} \cos 5x\,dx is
  39. Q381 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2013 · Paper 1If \frac{dy}{dx} = \cos x then
  40. Q391 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2013 · Paper 1If f''(x) = 6x, then given that f'(0) = 0, and c is a constant, f(x) =
  41. Q411 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2013 · Paper 1Given that \int_2^5 4\,f(x)\,dx = 9, the value of \int_2^5 3\,f(x)\,dx is
  42. Q441 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2013 · Paper 1Given \frac{dy}{dx} = 2x, then possible sketches of the graph of y are I. A parabola opening upwards with vertex at (0, 1), passing through (-1, 1) and (1, 1)? II. A parabola opening upwards with vertex at…
  43. Q351 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2014 · Paper 1Given that \cos 2x = 2\cos^2 x - 1, then \int \cos^2\left(\frac{x}{4}\right) dx is
  44. Q411 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2014 · Paper 1An expression for obtaining the volume generated by rotating the shaded region through 360^\circ about the x-axis is
  45. Q351 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2015 · Paper 1If \int_2^4 f(x)\,dx = 12, what is the value of \int_2^3 f(x)\,dx + \int_3^4 (f(x) - 1)\,dx?
  46. Q381 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2015 · Paper 1If the rate of change of y with respect to x is 3x^2 + \frac{4}{x^3}, then y is equal to
  47. Q391 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2015 · Paper 1If f''(x) = 6x then given that f'(0) = 0 and c is a constant, f(x) =
  48. Q401 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2015 · Paper 1If f'(x) = \sin x, then f(x) =
  49. Q411 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2015 · Paper 1At time t years, the growth of a certain country's gross national product, G, is given by the equation \frac{dG}{dt} = 5 + \cos t. At the beginning of the year 1990, the gross national product was recorded as…
  50. Q431 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2015 · Paper 1The TOTAL shaded area in the diagram below is given by
  51. 6(a)7 marks· CAPE Pure Mathematics Unit 1 · May/June 2023 · Paper 2Calculate the volume of the solid generated by revolving the region bounded by the line y = 3x - 6 and the parabola y = x^2 + 3x - 2, on the interval [0, 1] about the x-axis.
  52. 6(b)7 marks· CAPE Pure Mathematics Unit 1 · May/June 2023 · Paper 2The diagram below shows the curves y = \cos x and y = \sin x. Determine the area bounded by the curves between x = \pi/4 and x = 3\pi/2.
  53. 6(d)4 marks· CAPE Pure Mathematics Unit 1 · May/June 2023 · Paper 2Determine \int \cos^3 2x \sin 2x \, dx.
  54. 6(a)7 marks· CAPE Pure Mathematics Unit 1 · May/June 2024 · Paper 2Calculate the volume of the solid generated by revolving the region bounded by the graphs of x = sqrt(y) and y = 2x about the y-axis.
  55. 6(b)7 marks· CAPE Pure Mathematics Unit 1 · May/June 2024 · Paper 2Calculate the area between the line and the curve.
  56. 6(c)5 marks· CAPE Pure Mathematics Unit 1 · May/June 2024 · Paper 2Given that f is an even function, show that integral from -a to a of f(x) dx = 2 * integral from 0 to a of f(x) dx.
  57. 6(d)6 marks· CAPE Pure Mathematics Unit 1 · May/June 2024 · Paper 2Solve the differential equation x^3 y' = 2 - x^4, given that at x = 1, y = 2.
  58. 5(e)6 marks· CAPE Pure Mathematics Unit 1 · May/June 2026 · Paper 2Determine ∫₂⁴ x(x - 4)⁵ dx using the substitution u = x - 4.
  59. 6(b)6 marks· CAPE Pure Mathematics Unit 1 · May/June 2026 · Paper 2Calculate the area of the shaded region.
  60. 6(c)5 marks· CAPE Pure Mathematics Unit 1 · May/June 2026 · Paper 2The gradient at any point (x, y) on a curve is equal to (5x + 1)². Given that the curve passes through the point (1, 0.2), determine the equation of the curve.
  61. 5(a)6 marks· Pure Mathematics · Unit 1 Q5 5(a)Use the substitution u = x² + 2 to determine ∫(x² + 2)³ (4x³) dx.
  62. 5(a)6 marks· Pure Mathematics · Unit 1 Q5 5(a)Use the substitution u = (x³ + 4) to determine ∫ 3x² (x³ + 4)⁴ dx.
  63. 5(a)4 marks· Pure Mathematics · Unit 1 Q5 5(a)Use an appropriate substitution to find ∫x(x + 1)³ dx.
  64. 5(a)6 marks· Pure Mathematics · Unit 1 Q5 5(a)With the aid of a diagram, show that for n = 4, ∫₀ᵇ f(x)dx ≈ (d/2)[y₀ + 2y₁ + 2y₂ + 2y₃ + y₄], where y₀ = f(a), y₁ = f(a + d), ..., y₄ = f(a + 4d).
  65. 5(b)6 marks· Pure Mathematics · Unit 1 Q5 5(b)Use the Trapezium rule to find the approximate area of a flat side of each disc by using eight (8) subintervals.
  66. 5(b)5 marks· Pure Mathematics · Unit 1 Q5 5(b)Calculate the volume of the solid that results from rotating R about the y-axis.
  67. 5(b)6 marks· Pure Mathematics · Unit 1 Q5 5(b)Calculate the area of the region lying between the parabolas y = x² and x = (1/8)y².
  68. 5(b)9 marks· Pure Mathematics · Unit 1 Q5 5(b)Calculate the area of the region enclosed by the parabolas y = x² and y = 6x - 3x².
  69. 5(b)(i)8 marks· Pure Mathematics · Unit 1 Q5 5(b)(i)Write down an appropriate differential equation connecting the length of the rod and the rate of decrease at time, t. Hence, derive an expression for L.
  70. 5(b)(ii)7 marks· Pure Mathematics · Unit 1 Q5 5(b)(ii)Evaluate the length, L, of the rod using the trapezium rule for five ordinates and strips of length 1 unit.
  71. 5(b)(v)5 marks· Pure Mathematics · Unit 1 Q5 5(b)(v)Find the area bounded by the curve and the interval of the x-axis, -2 < x < 0.
  72. 5(c)6 marks· Pure Mathematics · Unit 1 Q5 5(c)Show that ∫₀¹ (eˣ / (eˣ + e¹⁻ˣ)) dx = 1/2.
  73. 5(c)4 marks· Pure Mathematics · Unit 1 Q5 5(c)Compare this approximate area with the exact area of a side of each disc.
  74. 5(c)(i)3 marks· Pure Mathematics · Unit 1 Q5 5(c)(i)Find the equation of C.
  75. 5(c)(i)6 marks· Pure Mathematics · Unit 1 Q5 5(c)(i)Evaluate ∫ from -1 to 1 of (x - 1/x)² dx.
  76. 5(c)(ii)4 marks· Pure Mathematics · Unit 1 Q5 5(c)(ii)Find the volume of the solid generated by rotating the shaded area through 2π radians about the x-axis.
  77. 5(c)(ii)4 marks· Pure Mathematics · Unit 1 Q5 5(c)(ii)Using the substitution u = x² + 4, or otherwise, find ∫ of x / √(x² + 4) dx.
  78. 5(d)5 marks· Pure Mathematics · Unit 1 Q5 5(d)Find the function f(x).
  79. 5(d)(i)2 marks· Pure Mathematics · Unit 1 Q5 5(d)(i)Sketch the solid that is generated by the rotation.
  80. 5(d)(ii)6 marks· Pure Mathematics · Unit 1 Q5 5(d)(ii)Find the volume of the solid that is generated.
  81. 5(d)(ii)5 marks· Pure Mathematics · Unit 1 Q5 5(d)(ii)Calculate the area of the shaded region.
  82. 6(a)6 marks· Pure Mathematics · Unit 1 Q6 6(a)Calculate the volume generated when the finite region in the first quadrant bounded by the curve, y = 2x², the y-axis and the line y = 2 is rotated completely about the y-axis.
  83. 6(a)6 marks· Pure Mathematics · Unit 1 Q6 6(a)Use the substitution u = 3x² + 1 to find ∫ x dx / √(3x² + 1).
  84. 6(a)6 marks· Pure Mathematics · Unit 1 Q6 6(a)Using the substitution u = x + 3 or otherwise, evaluate ∫ x√(x+3) dx.
  85. 6(a)(i)5 marks· Pure Mathematics · Unit 1 Q6 6(a)(i)By using the substitution u = 1 − x, find ∫ x (1-x)² dx.
  86. 6(a)(i)2 marks· Pure Mathematics · Unit 1 Q6 6(a)(i)Use the result ∫_0^a f(x)dx = ∫_0^a f(a-x)dx, a > 0, to show that if I = ∫_0^(π/2) sin^2 x dx, then I = ∫_0^(π/2) cos^2 x dx.
  87. 6(a)(i)3 marks· Pure Mathematics · Unit 1 Q6 6(a)(i)Determine the area of the region, A.
  88. 6(a)(i)a)4 marks· Pure Mathematics · Unit 1 Q6 6(a)(i)a)Find the equation of the curve
  89. 6(a)(ii)6 marks· Pure Mathematics · Unit 1 Q6 6(a)(ii)Hence, or otherwise, show that I = π/4.
  90. 6(a)(ii)4 marks· Pure Mathematics · Unit 1 Q6 6(a)(ii)show that ∫[f(t) + g(t)] dt = ∫f(t) dt + ∫g(t) dt.
  91. 6(a)(ii)6 marks· Pure Mathematics · Unit 1 Q6 6(a)(ii)Hence, use integration to determine the area bounded by the lines.
  92. 6(a)(ii)5 marks· Pure Mathematics · Unit 1 Q6 6(a)(ii)Calculate the volume of the solid that results from revolving the region, A, about the x-axis.
  93. 6(a)(iii)6 marks· Pure Mathematics · Unit 1 Q6 6(a)(iii)Find the area between the curve and the positive x-axis.
  94. 6(a)(iii)7 marks· Pure Mathematics · Unit 1 Q6 6(a)(iii)Hence, use integration to determine the area of triangle PQR.
  95. 6(b)6 marks· Pure Mathematics · Unit 1 Q6 6(b)Given that ∫₀ᵃ (x + 1) dx = 3 ∫₀ᵃ (x - 1) dx, a > 0, find the value of the constant a.
  96. 6(b)4 marks· Pure Mathematics · Unit 1 Q6 6(b)Find the equation of C.
  97. 6(b)7 marks· Pure Mathematics · Unit 1 Q6 6(b)Given that the curve passes through the point (1,1), determine an expression for f(x).
  98. 6(b)5 marks· Pure Mathematics · Unit 1 Q6 6(b)The voltage in a circuit, V, satisfies the equation dV/dt + V/2.5 = 0. Given that V = 25 volts when t = 0 seconds, write an expression for V in terms of t.
  99. 6(b)(i)5 marks· Pure Mathematics · Unit 1 Q6 6(b)(i)Calculate the area of R.
  100. 6(b)(i)8 marks· Pure Mathematics · Unit 1 Q6 6(b)(i)Find the volume generated by direct integration.
  101. 6(b)(i)3 marks· Pure Mathematics · Unit 1 Q6 6(b)(i)Determine the equation of the curve.
  102. 6(b)(i)4 marks· Pure Mathematics · Unit 1 Q6 6(b)(i)Find ∫ from 1 to 4 of [3 f(x) + 4] dx.
  103. 6(b)(i)3 marks· Pure Mathematics · Unit 1 Q6 6(b)(i)Given that ∫(from 1 to 6) f(x) dx = 7, evaluate ∫(from 1 to 6) [2 - f(x)] dx.
  104. 6(b)(i)2 marks· Pure Mathematics · Unit 1 Q6 6(b)(i)Use the result ∫(from 0 to a) f(x) dx = ∫(from 0 to a) f(a-x) dx, where a > 0, to show that ∫(from 0 to π) x sin x dx = ∫(from 0 to π) (π - x) sin x dx.
  105. 6(b)(i)5 marks· Pure Mathematics · Unit 1 Q6 6(b)(i)Evaluate ∫₀³ f(x) dx.
  106. 6(b)(ii)11 marks· Pure Mathematics · Unit 1 Q6 6(b)(ii)Find the volume generated by the trapezium rule, using five coordinates.
  107. 6(b)(ii)5 marks· Pure Mathematics · Unit 1 Q6 6(b)(ii)The area of R is estimated using the trapezium rule with 2 intervals of equal width. Show that this trapezium rule estimate differs by 1/3 from the exact value for the area of R found in (b)(i).
  108. 6(b)(ii)4 marks· Pure Mathematics · Unit 1 Q6 6(b)(ii)Hence, evaluate ∫₀² (12 - 3x²)/(x² + 4)² dx.
  109. 6(b)(ii)3 marks· Pure Mathematics · Unit 1 Q6 6(b)(ii)Hence, evaluate ∫ (from 0 to 1) (24-6x²)/(x²+4)² dx.
  110. 6(b)(ii)4 marks· Pure Mathematics · Unit 1 Q6 6(b)(ii)Find the value of the constant k.
  111. 6(b)(ii)4 marks· Pure Mathematics · Unit 1 Q6 6(b)(ii)Using the substitution u = x + 1, evaluate ∫ from 0 to 3 of 2f(x + 1) dx.
  112. 6(b)(ii)8 marks· Pure Mathematics · Unit 1 Q6 6(b)(ii)Calculate the volume created by rotating the plane figure bounded by x = 0, y = 4, y = 5 and the curve y = x^2 + 4 through 360° about the y-axis.
  113. 6(b)(ii)4 marks· Pure Mathematics · Unit 1 Q6 6(b)(ii)Find the volume generated by rotating the area bounded by the curve in (b) (i) above, the x-axis, and the lines x = 0 and x = 2 about the x-axis.
  114. 6(b)(ii)a)2 marks· Pure Mathematics · Unit 1 Q6 6(b)(ii)a)Show that ∫(from 0 to π) x sin x dx = ∫(from 0 to π) sin x dx - ∫(from 0 to π) x sin x dx
  115. 6(b)(ii)b)5 marks· Pure Mathematics · Unit 1 Q6 6(b)(ii)b)Show that ∫(from 0 to π) x sin x dx = π.
  116. 6(b)(iii)4 marks· Pure Mathematics · Unit 1 Q6 6(b)(iii)On a carefully labelled sketch of y = x² + 1, shade in the 2 trapezia which are used to estimate the area of R.
  117. 6(b)(iv)5 marks· Pure Mathematics · Unit 1 Q6 6(b)(iv)Another approximation for the area of R is obtained using 2 trapezia of unequal width. The first trapezium has width, h, and the second trapezium width, (2 - h) with the three ordinates occurring where x = 0, x = h and…
  118. 6(c)5 marks· Pure Mathematics · Unit 1 Q6 6(c)By using the substitution, y = x², show that ∫ 1/(√y + √y³) dy = ∫ 2/(1 + x²) dx.
  119. 6(c)4 marks· Pure Mathematics · Unit 1 Q6 6(c)Calculate ∫₁² [f(x) + g(x)] dx.
  120. 6(c)5 marks· Pure Mathematics · Unit 1 Q6 6(c)Find the value of u > 0 if ∫ (from 1 to u) (2/x⁴) dx = 7/192.
  121. 6(c)5 marks· Pure Mathematics · Unit 1 Q6 6(c)Given that ∫[-1,3] (3f(x) + g(x)) dx = 5 and ∫[-1,3] (5f(x) - 2g(x)) dx = 1, determine ∫[-1,3] f(x) dx and ∫[-1,3] g(x) dx.
  122. 6(c)(i)4 marks· Pure Mathematics · Unit 1 Q6 6(c)(i)Use the substitution t = a - x to show that ∫₀ᵃ f(x) dx = ∫₀ᵃ f(a-x) dx.
  123. 6(c)(i)5 marks· Pure Mathematics · Unit 1 Q6 6(c)(i)Formulate an appropriate differential equation and find the equation of the curve family.
  124. 6(c)(ii)9 marks· Pure Mathematics · Unit 1 Q6 6(c)(ii)Hence find the exact value of the area of the shaded region.
  125. 6(c)(ii)7 marks· Pure Mathematics · Unit 1 Q6 6(c)(ii)Find the volume obtained by rotating the portion of the curve between x = 0 and x = 1 through 2π radians about the y axis.
  126. 6(c)(ii)13 marks· Pure Mathematics · Unit 1 Q6 6(c)(ii)Hence, or otherwise, determine the equation of the curve given that it passes through the point (1, 3).
  127. 6(c)(ii)3 marks· Pure Mathematics · Unit 1 Q6 6(c)(ii)If ∫₀⁴ f(x) dx = 12, use the substitution t = x - 1 to evaluate ∫₁⁵ 3f(x-1) dx.
  128. 6(c)(ii)5 marks· Pure Mathematics · Unit 1 Q6 6(c)(ii)Calculate the area of the shaded portion of the diagram bounded by the curve and the straight line.
  129. 6(d)5 marks· Pure Mathematics · Unit 1 Q6 6(d)find ∫sin²x cos²x dx.
  130. 6(d)(i)6 marks· Pure Mathematics · Unit 1 Q6 6(d)(i)Solve the differential equation dy/dx = sin x / sin y given that when x = 0, y = π/2.
  131. 6(d)(ii)4 marks· Pure Mathematics · Unit 1 Q6 6(d)(ii)Determine the equation of the curve that passes through (1, 5) and for which y = ∫ 6x² dx.
  132. 6(e)6 marks· Pure Mathematics · Unit 1 Q6 6(e)Find the area bounded by this curve and the x-axis for -2 ≤ x ≤ 1.
  133. 6(e)5 marks· Pure Mathematics · Unit 1 Q6 6(e)Find the total area enclosed between the curve, the x-axis and the values x = -2 and x = 2.

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