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Integral calculus: board exam questions with solutions

On the ECE exam under Mathematics and the REE exam under Mathematics. Try each question before you open its solution, then practise with fresh numbers below: 2 computed question types on this topic.

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Questions

1. Evaluate the integral of 3x² dx from x = 0 to x = 2.

  1. 8
  2. 12
  3. 6
  4. 24
Show the answer and solution

Answer: A. 8

The antiderivative is x³, and 2³ − 0 = 8.

2. Find ∫ cos x dx.

  1. −sin x + C
  2. cos x + C
  3. −cos x + C
  4. sin x + C
Show the answer and solution

Answer: D. sin x + C

d/dx (sin x) = cos x, so ∫ cos x dx = sin x + C.

3. Find the area enclosed between y = x and y = x² from x = 0 to x = 1.

  1. 1/2
  2. 1/6
  3. 1/3
  4. 5/6
Show the answer and solution

Answer: B. 1/6

For 0 ≤ x ≤ 1, x ≥ x², so A = ∫(x − x²)dx = 1/2 − 1/3 = 1/6.

4. Evaluate the integral of sin x dx from 0 to π.

  1. 2
  2. 0
  3. 1
  4. −2
Show the answer and solution

Answer: A. 2

[−cos x] from 0 to π = −cos π + cos 0 = 1 + 1 = 2.

5. Find ∫ x eˣ dx.

  1. eˣ(x + 1) + C
  2. x²eˣ/2 + C
  3. eˣ(x − 1) + C
  4. xeˣ + C
Show the answer and solution

Answer: C. eˣ(x − 1) + C

Integrate by parts with u = x and dv = eˣdx: xeˣ − ∫eˣdx = eˣ(x − 1) + C.

6. What is the average value of f(x) = x² on the interval [0, 3]?

  1. 9
  2. 4.5
  3. 1
  4. 3
Show the answer and solution

Answer: D. 3

Average = (1/3)∫₀³ x² dx = (1/3)(27/3) = (1/3)(9) = 3.

7. The region under y = √x from x = 0 to x = 4 is revolved about the x-axis. Find the volume of the solid generated.

  1. 8.00 cubic units
  2. 50.27 cubic units
  3. 25.13 cubic units
  4. 16.76 cubic units
Show the answer and solution

Answer: C. 25.13 cubic units

By the disk method, V = π∫₀⁴ (√x)² dx = π∫₀⁴ x dx = π(16/2) = 8π = 25.13 cubic units.

8. Using the integral definition of RMS, V_rms = √[(1/T)∫v² dt], find the RMS value of v = 100 sin ωt V.

  1. 63.66 V
  2. 70.71 V
  3. 50.00 V
  4. 100.00 V
Show the answer and solution

Answer: B. 70.71 V

(1/T)∫(100 sin ωt)² dt = 100²/2 = 5000, so V_rms = √5000 = 70.71 V. 63.66 V is the average over a half cycle.

Worked examples with computed numbers

These numbers are fixed on this page; in practice they change every time.

9. Evaluate ∫ from 2 to 6 of (4x² − 2x) dx.

  1. 258.7
  2. 252
  3. 120
  4. 245.3
Show the answer and solution

Answer: D. 245.3

∫(4x² − 2x)dx = 4x³/3 − 2x²/2; from 2 to 6: 245.3.

10. Find the area bounded by y = x² and y = 9x.

  1. 121.5
  2. 40.5
  3. 364.5
  4. 243
Show the answer and solution

Answer: A. 121.5

They meet at x = 0 and x = 9. A = ∫₀^9 (9x − x²)dx = 9³/2 − 9³/3 = 9³/6 = 121.5.

Other topics in the same subjects

Algebra · Trigonometry · Analytic geometry · Differential calculus · Differential equations · Complex numbers · Probability and statistics · Matrices and determinants · Laplace transforms and Fourier series

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