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Integration practice

QCE Mathematical Methods · Original practice questions with worked solutions

All integration practice questions

73 original questions · Page 3 of 4

  1. Q252 · Original practice · 1 mark
    Values of ff are f(0)=2f(0)=2, f(1)=5f(1)=5 and f(2)=4f(2)=4. The trapezoidal estimate of ∫02f(x) dx\int_0^2 f(x)\,dx with two equal strips is
    Further integration
  2. Q281 · Original practice · 4 marks
    A function satisfies f′(x)=6x2+4e2x−3xf^{\prime}(x)=6x^2+4e^{2x}-\dfrac3x for x>0x>0 and f(1)=2e2f(1)=2e^2.
    Introduction to integration
  3. Q282 · Original practice · 6 marks
    A delivery robot moves along a straight corridor with acceleration a(t)=6t−4 m/s2a(t)=6t-4\ \mathrm{m/s^2}. At t=0t=0, its velocity is 1 m/s1\ \mathrm{m/s} and its position is 22 m from the reference point.
    Introduction to integration
  4. Q283 · Original practice · 6 marks
    A small cart has velocity v(t)=et−2 m/sv(t)=e^t-2\ \mathrm{m/s} for 0≤t≤ln⁡40\leq t\leq\ln4.
    Further integration
  5. Q284 · Original practice · 4 marks
    The growth rate of a vine is h′(t)=12t+2h^{\prime}(t)=\dfrac{12}{t+2} cm per week, for t≥0t\geq0. Its initial height is 1515 cm.
    Introduction to integration
  6. Q289 · Original practice · 4 marks
    Evaluate the following integrals exactly.
    Further integration
  7. Q290 · Original practice · 5 marks
    The graph of y=x2−1y=x^2-1 is shown for −2≤x≤2-2\leq x\leq2.
    Further integration
  8. Q291 · Original practice · 5 marks
    The curves y=x2y=x^2 and y=x+2y=x+2 enclose a finite region.
    Further integration
  9. Q292 · Original practice · 5 marks
    A light sculpture has vertical profile y=2sin⁡x−1y=2\sin x-1 for 0≤x≤π0\leq x\leq\pi.
    Further integration
  10. Q293 · Original practice · 4 marks
    A mural has a curved upper edge above a horizontal baseline. Its measured heights hh metres at distances xx metres are shown: x01234h1.22.12.82.01.3\begin{array}{c|ccccc}x&0&1&2&3&4\\\hline h&1.2&2.1&2.8&2.0&1.3\end{array}.
    Further integration
  11. Q294 · Original practice · 5 marks
    The marginal production cost of xx souvenir pins is modelled by C′(x)=0.04x+2+20x+10C^{\prime}(x)=0.04x+2+\dfrac{20}{x+10} dollars per pin, for x≥0x\geq0. The fixed cost is C(0)=50C(0)=50 dollars.
    Further integration
  12. Q295 · Original practice · 6 marks
    Water enters a tank at rate I(t)=8+4sin⁡(πt/6)I(t)=8+4\sin(\pi t/6) litres per minute, while water leaves at 66 litres per minute. Initially the tank contains 2020 litres. The model applies for 0≤t≤120\leq t\leq12.
    Further integration
  13. Q296 · Original practice · 4 marks
    Let F(x)=∫0x2(1+sin⁡t) dtF(x)=\displaystyle\int_0^{x^2}(1+\sin t)\,dt.
    Further integration
  14. Q323 · Original practice · 8 marks
    Visitors arrive at a night market at rate R(t)=120(e−t/2−e−t)R(t)=120(e^{-t/2}-e^{-t}) people per hour for t≥0t\geq0. Let N(t)N(t) be the modelled number who have arrived by time tt, with N(0)=0N(0)=0.
    Further integration
  15. Q331 · Original practice · 8 marks
    A camera slider has velocity v(t)=2cos⁡t−e−tv(t)=2\cos t-e^{-t} metres per second for 0≤t≤π0\leq t\leq\pi. Its initial position is s(0)=0s(0)=0.
    Further integration
  16. Q333 · Original practice · 7 marks
    A curved garden border has height f(x)=5+cos⁡(πx/4)f(x)=5+\cos(\pi x/4) metres for 0≤x≤40\leq x\leq4.
    Further integration
  17. Q337 · Original practice · 8 marks
    A robot's position is s(t)=t3−6t2+9ts(t)=t^3-6t^2+9t metres for 0≤t≤40\leq t\leq4 seconds. An inspection occurs at a random time TT uniformly distributed on [0,4][0,4].
    Further integration
  18. Q345 · Original practice · 1 mark
    The shaded rectangles use left endpoints and four equal strips to estimate ∫04e0.2x dx\int_0^4 e^{0.2x}\,dx. Determine the estimate to three decimal places.
    Further integration
  19. Q346 · Original practice · 1 mark
    Use the trapezoidal rule with six equal strips to estimate ∫031x+1 dx\displaystyle\int_0^3\frac{1}{x+1}\,dx. Give the estimate to three decimal places.
    Further integration
  20. Q359 · Original practice · 4 marks
    f′(x)=3cos⁡(2x)−2e−xf^{\prime}(x)=3\cos(2x)-2e^{-x} and f(π)=4f(\pi)=4. Determine f(x)f(x) exactly.
    Introduction to integration
  21. Q360 · Original practice · 5 marks
    A particle has acceleration a(t)=2cos⁡ta(t)=2\cos t m/s2^2 for t≥0t\geq0. Its velocity at t=0t=0 is 11 m/s and its position at t=π/2t=\pi/2 is 33 m.
    Introduction to integration
  22. Q361 · Original practice · 5 marks
    Two particles move on a straight line with velocities vA(t)=t2−2t+2v_A(t)=t^2-2t+2 and vB(t)=2t+2v_B(t)=2t+2 m/s for t≥0t\geq0. They have equal velocities at t=0t=0.
    Introduction to integration
  23. Q368 · Original practice · 5 marks
    The curve is y=x2+1y=x^2+1 for 0≤x≤20\leq x\leq2.
    Further integration
  24. Q369 · Original practice · 3 marks
    It is known that ∫15f(x) dx=12\int_1^5 f(x)\,dx=12 and ∫13f(x) dx=7\int_1^3 f(x)\,dx=7.
    Further integration