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Integration practice

QCE Mathematical Methods · Original practice questions with worked solutions

All integration practice questions

73 original questions · Page 2 of 4

  1. Q120 · Original practice · 5 marks
    A twice-differentiable function satisfies f′′(x)=6x−4f''(x)=6x-4, f′(1)=0f'(1)=0 and ∫02f(x) dx=10\displaystyle\int_0^2 f(x)\,dx=10. Determine f(0)f(0).
    Introduction to integration
  2. Q126 · Original practice · 1 mark
    Evaluate ∫0π/2cos⁡x dx\displaystyle\int_0^{\pi/2}\cos x\,dx.
    Further integration
  3. Q127 · Original practice · 1 mark
    The values of a function are shown graphically at x=0,1,2,3x=0,1,2,3. Using the trapezoidal rule with width 11, estimate ∫03f(x) dx\displaystyle\int_0^3 f(x)\,dx.
    Further integration
  4. Q128 · Original practice · 3 marks
    Determine the area enclosed by the curve y=x(4−x)y=x(4-x) and the xx-axis.
    Further integration
  5. Q129 · Original practice · 4 marks
    The line y=3xy=3x and the parabola y=x2+2y=x^2+2 enclose a bounded region. Determine its area.
    Further integration
  6. Q130 · Original practice · 6 marks
    A tank initially contains V0V_0 litres of water. For 0≤t≤100\le t\le10 minutes, water enters at a rate 4e−0.2t4e^{-0.2t} L/min while a pump removes water at a constant rate of 22 L/min. Determine the minimum value of V0V_0 that guarantees the tank does not become empty during the 10-minute interval.
    Further integration
  7. Q166 · Original practice · 1 mark
    An antiderivative of 4ex−3sin⁡x4e^x-3\sin x is
    Introduction to integration
  8. Q167 · Original practice · 1 mark
    A curve has gradient dydx=4−3x2\dfrac{dy}{dx}=4-3x^2 and passes through (1,5)(1,5). Determine its yy-intercept.
    Introduction to integration
  9. Q168 · Original practice · 1 mark
    A particle has acceleration a(t)=4ta(t)=4t m/s2^2 and initial velocity v(0)=2v(0)=2 m/s. Determine v(3)v(3).
    Introduction to integration
  10. Q169 · Original practice · 5 marks
    The velocity–time graph of a particle is shown. The graph is linear between consecutive plotted points, and the initial position is s(0)=7s(0)=7 m.
    Determine (a) the displacement from t=0t=0 to t=6t=6, (b) the total distance travelled, and (c) the position at t=6t=6.
    Introduction to integration
  11. Q170 · Original practice · 5 marks
    A particle moves on a straight line with acceleration a(t)=6e−ta(t)=6e^{-t} m/s2^2, initial velocity v(0)=−1v(0)=-1 m/s and initial position s(0)=2s(0)=2 m. Determine the first time after t=0t=0 at which the particle is at rest, and determine its position at that time.
    Introduction to integration
  12. Q176 · Original practice · 1 mark
    Given ∫02f(x) dx=5\displaystyle\int_0^2 f(x)\,dx=5 and ∫25f(x) dx=−1\displaystyle\int_2^5 f(x)\,dx=-1, determine ∫05f(x) dx\displaystyle\int_0^5 f(x)\,dx.
    Further integration
  13. Q177 · Original practice · 1 mark
    The area under the line y=4−xy=4-x and above the xx-axis for 0≤x≤40\le x\le4 is
    Further integration
  14. Q178 · Original practice · 2 marks
    The values f(0)=2f(0)=2, f(1)=3f(1)=3, f(2)=5f(2)=5, f(3)=4f(3)=4 and f(4)=2f(4)=2 are given. Use the trapezoidal rule with width 11 to estimate ∫04f(x) dx\displaystyle\int_0^4 f(x)\,dx.
    Further integration
  15. Q179 · Original practice · 4 marks
    The graphs of y=2xy=2x and y=x2y=x^2 are shown. Determine the area of the bounded region between the curves.
    Further integration
  16. Q180 · Original practice · 5 marks
    A tank contains 8080 L at t=0t=0. For 0≤t≤60\le t\le6 minutes, its volume changes at the rate dVdt=12−12t2\dfrac{dV}{dt}=12-\dfrac12t^2 litres per minute. Determine the time at which the volume is greatest, the greatest volume, and the volume at t=6t=6.
    Further integration
  17. Q203 · Original practice · 1 mark
    The exact area under y=3x2y=3x^{2} from x=0x=0 to x=2x=2 is
    Introduction to integration
  18. Q213 · Original practice · 5 marks
    The curves y=6−0.5x2y=6-0.5x^{2} and y=x+2y=x+2 enclose a finite region.
    Further integration
  19. Q226 · Original practice · 1 mark
    ∫02e2xdx\int_0^{2} e^{2x} \mathrm{d}x equals
    Further integration
  20. Q233 · Original practice · 5 marks
    The rate of water flow into a tank is R(t)=5+2sin⁡(πt6)Lmin−1R(t)=5+2\sin (\frac{\pi t}{6}) L min^{-1} for 0≤t≤120\le t\le 12.
    Further integration
  21. Q245 · Original practice · 1 mark
    Which is an antiderivative of 6cos⁡(2x)6\cos(2x)?
    Introduction to integration
  22. Q246 · Original practice · 1 mark
    A moving tile has velocity v(t)=4t−6v(t)=4t-6 and position s(0)=5s(0)=5. Which expression gives its position?
    Introduction to integration
  23. Q250 · Original practice · 1 mark
    The shaded region lies between y=3−xy=3-x and the xx-axis for 0≤x≤30\leq x\leq3. Its area is
    Further integration
  24. Q251 · Original practice · 1 mark
    The graph shows a velocity function. The displacement over 0≤t≤40\leq t\leq4 is ∫04(t−2) dt\int_0^4(t-2)\,dt. What is the total distance travelled?
    Further integration