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DSE M1 Applications of Definite Integrals Practice

DSE M1 · 60 practice questions with instant marking and worked solutions.

Applications of Definite Integrals is an important part of the DSE M1 mathematics curriculum. This topic develops area under a curve, area between curves, the trapezoidal rule and displacement from velocity. Begin by identifying what every number, symbol, diagram and unit represents. Translate the information into a mathematical relationship before calculating. Each line of working should have a purpose. Finally use estimation, substitution, an inverse operation, or a second representation to check the result. This habit separates a misunderstanding from a copying error and makes correction more useful.

A reliable approach to Applications of Definite Integrals has four stages. First, mark the given information and the quantity required. Second, choose an appropriate representation, property or formula. Third, carry out the operations in a clear order while retaining signs and units. Fourth, return to the original context and decide whether the result is possible. For a graph or table, read the title, scale and labels before using a value. For algebra, inspect brackets, negative signs and the order of operations before simplifying.

The 60 original questions in this topic use a shared mathematical specification in Chinese and English. Difficulty rises from level 1 to level 5. Early questions establish the language and direct skills of Applications of Definite Integrals; middle questions add two-step reasoning, conversions and common variations; later questions connect ideas and require a deliberate check. In multiple-choice work, calculate before looking at the choices. In fill-in and short-answer work, retain decisive steps. For true-or-false statements, support the decision with a calculation or counterexample. 12 of them are DSE Extended Part style structured questions answered in parts (a), (b), (c), where later parts build on earlier ones.

Effective revision is more than recording a score. After completing the Applications of Definite Integrals set, group errors by concept tags and decide whether each problem arose from interpretation, forming an expression, calculation, or presenting the answer. Retry the item on a later day without opening the solution, then test the same method with a different value. If the same tag appears repeatedly, return to the explanation and rebuild the concept. If signs or units are the issue, use a fixed pre-submission checklist.

Applications of Definite Integrals is closely linked with the rest of the Extended Part: algebraic and trigonometric skills underpin calculus, and calculus in turn supports probability distributions and statistical inference. A learner who can explain aloud why every step is valid has moved beyond imitation. Estimate a reasonable range, complete an exact solution, and compare the two. Agreement between context, estimate and exact answer is strong evidence that the solution is sound.

This batch adds 20 questions, bringing the topic to 60. The new questions cover controlled variations of different concepts, with paired bilingual conditions, hints, worked solutions and reproducible checks.

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  1. 1 Basic
    The curve $y=x^2$ and the line $y=3x$ enclose a region. Find the area of the region (as a fraction in lowest terms). Choose the correct answer.
  2. 2 Basic
    Use the trapezoidal rule with 2 sub-intervals to estimate $\int_0^{4}\frac{6}{x+1}\,dx$. Find the estimate (as a fraction in lowest terms). Give the answer only.

  3. 3 Basic
    A particle has velocity $v=4t+5$ m/s at time $t$ s. Find its displacement (in m) from $t=0$ to $t=2$. Show the essential working and give the answer.

  4. 4 Basic
    The value of $\int_{0}^{3}(-3x^{2}-6x+2)\,dx$ is 48. Is this true?
  5. 5 Basic
    Use the trapezoidal rule with 4 sub-intervals to estimate $\int_0^{4}\frac{12}{x+1}\,dx$. Find the estimate (as a fraction in lowest terms). Choose the correct answer.
  6. 6 Basic
    Use the trapezoidal rule with 2 sub-intervals to estimate $\int_0^{2}\frac{12}{x+1}\,dx$. Find the estimate. Give the answer only.

  7. 7 Basic
    A particle has velocity $v=3t+6$ m/s at time $t$ s. Find its displacement (in m) from $t=0$ to $t=6$. Show the essential working and give the answer.

  8. 8 Basic
    The value of $\int_{-2}^{2}(x^{2}+4x-2)\,dx$ is 8/3. Is this true?
  9. 9 Basic
    The curve $y=x^2$ and the line $y=6x$ enclose a region. Find the area of the region. Choose the correct answer.
  10. 10 Basic
    Use the trapezoidal rule with 2 sub-intervals to estimate $\int_0^{4}\frac{24}{x+1}\,dx$. Find the estimate (as a fraction in lowest terms). Give the answer only.

  11. 11 Basic
    A particle has velocity $v=t+6$ m/s at time $t$ s. Find its displacement (in m) from $t=0$ to $t=4$. Show the essential working and give the answer.

  12. 12 Basic
    The value of $\int_{-2}^{3}(4x^{2}-x+2)\,dx$ is -325/6. Is this true?
  13. 13 Standard
    The curve $y=x^2$ and the line $y=7x$ enclose a region. Find the area of the region (as a fraction in lowest terms). Choose the correct answer.
  14. 14 Standard
    Use the trapezoidal rule with 4 sub-intervals to estimate $\int_0^{2}\frac{60}{x+1}\,dx$. Find the estimate. Give the answer only.

  15. 15 Standard
    A particle has velocity $v=4t+5$ m/s at time $t$ s. Find its displacement (in m) from $t=0$ to $t=6$. Show the essential working and give the answer.

  16. 16 Standard DSE long question
    The curve $y=5x-x^2$ and the $x$-axis enclose a region. Answer each part; give the answer only.
    1. (a) Find the $x$-coordinate of the other $x$-intercept.
    2. (b) Find the greatest height of the curve (as a fraction in lowest terms).
    3. (c) Find the area of the region (as a fraction in lowest terms).
  17. 17 Standard
    Use the trapezoidal rule with 4 sub-intervals to estimate $\int_0^{4}\frac{6}{x+1}\,dx$. Find the estimate (as a fraction in lowest terms). Choose the correct answer.
  18. 18 Standard
    Use the trapezoidal rule with 2 sub-intervals to estimate $\int_0^{4}\frac{60}{x+1}\,dx$. Find the estimate. Give the answer only.

  19. 19 Standard
    A particle has velocity $v=t+1$ m/s at time $t$ s. Find its displacement (in m) from $t=0$ to $t=5$ (as a fraction in lowest terms). Show the essential working and give the answer.

  20. 20 Standard DSE long question
    The curve $y=9x-x^2$ and the $x$-axis enclose a region. Answer each part; give the answer only.
    1. (a) Find the $x$-coordinate of the other $x$-intercept.
    2. (b) Find the greatest height of the curve (as a fraction in lowest terms).
    3. (c) Find the area of the region (as a fraction in lowest terms).
  21. 21 Standard
    Use the trapezoidal rule with 2 sub-intervals to estimate $\int_0^{2}\frac{60}{x+1}\,dx$. Find the estimate. Choose the correct answer.
  22. 22 Standard
    Use the trapezoidal rule with 4 sub-intervals to estimate $\int_0^{4}\frac{24}{x+1}\,dx$. Find the estimate (as a fraction in lowest terms). Give the answer only.

  23. 23 Standard
    A particle has velocity $v=t+2$ m/s at time $t$ s. Find its displacement (in m) from $t=0$ to $t=5$ (as a fraction in lowest terms). Show the essential working and give the answer.

  24. 24 Standard DSE long question
    The curve $y=8x-x^2$ and the $x$-axis enclose a region. Answer each part; give the answer only.
    1. (a) Find the $x$-coordinate of the other $x$-intercept.
    2. (b) Find the greatest height of the curve.
    3. (c) Find the area of the region (as a fraction in lowest terms).
  25. 25 Advanced
    Find the value of $\int_{0}^{4}(-2x^{2}+5x+4)\,dx$ (as a fraction in lowest terms). Choose the correct answer.
  26. 26 Advanced
    Use the trapezoidal rule with 2 sub-intervals to estimate $\int_0^{2}\frac{24}{x+1}\,dx$. Find the estimate. Give the answer only.

  27. 27 Advanced
    Use the trapezoidal rule with 2 sub-intervals to estimate $\int_0^{4}\frac{12}{x+1}\,dx$. The estimate is 112/5. Is this true?
  28. 28 Advanced DSE long question
    Let $f(x)=12x^{2}-8x+5$. Answer each part; give the answer only.
    1. (a) Find $\int_0^1 f(x)\,dx$.
    2. (b) Find $\int_0^2 f(x)\,dx$.
    3. (c) If $F'(x)=f(x)$ and $F(0)=-4$, find $F(1)$.
  29. 29 Advanced
    Find the value of $\int_{0}^{2}(-2x^{2}+2)\,dx$ (as a fraction in lowest terms). Choose the correct answer.
  30. 30 Advanced
    The curve $y=x^2$ and the line $y=7x$ enclose a region. Find the area of the region (as a fraction in lowest terms). Give the answer only.

  31. 31 Advanced
    The value of $\int_{0}^{2}(x^{2}+2x+1)\,dx$ is 26/3. Is this true?
  32. 32 Advanced DSE long question
    Let $f(x)=3x^{2}-2x+1$. Answer each part; give the answer only.
    1. (a) Find $\int_0^1 f(x)\,dx$.
    2. (b) Find $\int_0^2 f(x)\,dx$.
    3. (c) If $F'(x)=f(x)$ and $F(0)=2$, find $F(1)$.
  33. 33 Advanced
    Find the value of $\int_{1}^{2}(4x^{2}+2x+1)\,dx$ (as a fraction in lowest terms). Choose the correct answer.
  34. 34 Advanced
    Use the trapezoidal rule with 2 sub-intervals to estimate $\int_0^{2}\frac{6}{x+1}\,dx$. Find the estimate. Give the answer only.

  35. 35 Advanced
    Use the trapezoidal rule with 4 sub-intervals to estimate $\int_0^{2}\frac{6}{x+1}\,dx$. The estimate is 67/10. Is this true?
  36. 36 Advanced DSE long question
    Let $f(x)=12x^{2}-2x-1$. Answer each part; give the answer only.
    1. (a) Find $\int_0^1 f(x)\,dx$.
    2. (b) Find $\int_0^2 f(x)\,dx$.
    3. (c) If $F'(x)=f(x)$ and $F(0)=2$, find $F(1)$.
  37. 37 Challenging
    A particle has velocity $v=3t+5$ m/s at time $t$ s. Find its displacement (in m) from $t=0$ to $t=3$ (as a fraction in lowest terms). Choose the correct answer.
  38. 38 Challenging
    Find the value of $\int_{-2}^{3}(3x^{2}-5x+6)\,dx$ (as a fraction in lowest terms). Show the essential working and give the answer.

  39. 39 Challenging
    Use the trapezoidal rule with 4 sub-intervals to estimate $\int_0^{4}\frac{6}{x+1}\,dx$. The estimate is 137/10. Is this true?
  40. 40 Challenging DSE long question
    The curve $y=6x-x^2$ and the $x$-axis enclose a region. Answer each part; give the answer only.
    1. (a) Find the $x$-coordinate of the other $x$-intercept.
    2. (b) Find the greatest height of the curve.
    3. (c) Find the area of the region.
  41. 41 Challenging
    A particle has velocity $v=4t+6$ m/s at time $t$ s. Find its displacement (in m) from $t=0$ to $t=4$. Choose the correct answer.
  42. 42 Challenging
    Find the value of $\int_{-1}^{4}(x^{2}+3x+5)\,dx$ (as a fraction in lowest terms). Show the essential working and give the answer.

  43. 43 Challenging
    The curve $y=x^2$ and the line $y=2x$ enclose a region. The area of the region is 8/3. Is this true?
  44. 44 Challenging DSE long question
    Let $f(x)=9x^{2}+8x+3$. Answer each part; give the answer only.
    1. (a) Find $\int_0^1 f(x)\,dx$.
    2. (b) Find $\int_0^2 f(x)\,dx$.
    3. (c) If $F'(x)=f(x)$ and $F(0)=4$, find $F(1)$.
  45. 45 Challenging
    A particle has velocity $v=t+3$ m/s at time $t$ s. Find its displacement (in m) from $t=0$ to $t=3$ (as a fraction in lowest terms). Choose the correct answer.
  46. 46 Challenging
    Find the value of $\int_{-2}^{2}(2x^{2}+x+6)\,dx$ (as a fraction in lowest terms). Show the essential working and give the answer.

  47. 47 Challenging
    The curve $y=x^2$ and the line $y=3x$ enclose a region. The area of the region is 9. Is this true?
  48. 48 Challenging DSE long question
    The curve $y=10x-x^2$ and the $x$-axis enclose a region. Answer each part; give the answer only.
    1. (a) Find the $x$-coordinate of the other $x$-intercept.
    2. (b) Find the greatest height of the curve.
    3. (c) Find the area of the region (as a fraction in lowest terms).
  49. 49 Elite
    Use the trapezoidal rule with 4 sub-intervals to estimate $\int_0^{4}\frac{12}{x+1}\,dx$. Find the estimate (as a fraction in lowest terms). Give the answer only.

  50. 50 Elite
    A particle has velocity $v=3t+5$ m/s at time $t$ s. Find its displacement (in m) from $t=0$ to $t=6$. Show the essential working and give the answer.

  51. 51 Elite
    The value of $\int_{0}^{4}(2x^{2}+x-3)\,dx$ is 116/3. Is this true?
  52. 52 Elite DSE long question
    Let $f(x)=12x^{2}-6$. Answer each part; give the answer only.
    1. (a) Find $\int_0^1 f(x)\,dx$.
    2. (b) Find $\int_0^2 f(x)\,dx$.
    3. (c) If $F'(x)=f(x)$ and $F(0)=-4$, find $F(1)$.
  53. 53 Elite
    A particle has velocity $v=t$ m/s at time $t$ s. Find its displacement (in m) from $t=0$ to $t=4$. Give the answer only.

  54. 54 Elite
    A particle has velocity $v=4t+1$ m/s at time $t$ s. Find its displacement (in m) from $t=0$ to $t=6$. Show the essential working and give the answer.

  55. 55 Elite
    The value of $\int_{1}^{4}(2x^{2}-3x+5)\,dx$ is 69/2. Is this true?
  56. 56 Elite DSE long question
    Let $f(x)=12x^{2}+1$. Answer each part; give the answer only.
    1. (a) Find $\int_0^1 f(x)\,dx$.
    2. (b) Find $\int_0^2 f(x)\,dx$.
    3. (c) If $F'(x)=f(x)$ and $F(0)=-4$, find $F(1)$.
  57. 57 Elite
    Use the trapezoidal rule with 2 sub-intervals to estimate $\int_0^{4}\frac{12}{x+1}\,dx$. Find the estimate (as a fraction in lowest terms). Give the answer only.

  58. 58 Elite
    A particle has velocity $v=2t+5$ m/s at time $t$ s. Find its displacement (in m) from $t=0$ to $t=5$. Show the essential working and give the answer.

  59. 59 Elite
    The value of $\int_{1}^{4}(-x^{2}+x+3)\,dx$ is -9/2. Is this true?
  60. 60 Elite DSE long question
    Let $f(x)=3x^{2}-8x-3$. Answer each part; give the answer only.
    1. (a) Find $\int_0^1 f(x)\,dx$.
    2. (b) Find $\int_0^2 f(x)\,dx$.
    3. (c) If $F'(x)=f(x)$ and $F(0)=1$, find $F(1)$.

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Common mistakes

Common mistakes in Applications of Definite Integrals

This topic needs particular care with area under a curve, area between curves, the trapezoidal rule and displacement from velocity. The first mistake is applying a remembered rule before reading all conditions. The second is compressing working so much that a sign, bracket, decimal point or unit disappears. The third is accepting an impossible answer without checking: for example, a probability above one, a negative physical length, an angle outside the geometric constraints, or a result far from the estimate. Separate the conditions, model, calculation and check, and finish by stating what the answer represents.

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