10 questions · Form 5 Additional Mathematics Bab 3: Integration
10 questions · plays in fullscreen
Prefer reading to quizzing? All 10 questions are listed below with the answer and explanation under each one.
1. If d/dx [f(x)] = 3x² + 2x, find f(x) given that f(1) = 4.
Answer: A
f(x) = ∫ (3x² + 2x) dx = x³ + x² + c. f(1) = 1³ + 1² + c = 4 => 2 + c = 4 => c = 2. f(x) = x³ + x² + 2.
2. Find the indefinite integral ∫ (3x² - 4x + 5) dx.
Answer: A
∫ (3x² - 4x + 5) dx = 3(x³/3) - 4(x²/2) + 5x + c = x³ - 2x² + 5x + c.
3. Find the volume of revolution generated when the region bounded by the curve y = √x, the x-axis, and x = 4 is rotated 360° about the x-axis.
Answer: A
V = π ∫₀⁴ y² dx = π ∫₀⁴ (√x)² dx = π ∫₀⁴ x dx = π [x²/2]₀⁴ = π (162 - 0) = 8π units³.
4. Evaluate the definite integral ∫₁³ (3x²) dx.
Answer: A
∫₁³ (3x²) dx = [x³]₁³ = (3³) - (1³) = 27 - 1 = 26.
5. Given that ∫₂⁵ g(x) dx = 6 and ∫₅⁸ g(x) dx = -2, find the value of ∫₂⁸ g(x) dx.
Answer: C
Using property: ∫₂⁸ g(x) dx = ∫₂⁵ g(x) dx + ∫₅⁸ g(x) dx = 6 + (-2) = 4.
6. Find the area of the region bounded by the curve y = x², the x-axis, and the lines x = 0 and x = 3.
Answer: C
Area = ∫₀³ x² dx = [x³/3]₀³ = (273) - 0 = 9 units².
7. Which integral gives the volume generated when the curve x = y² from y = 1 to y = 3 is rotated 360° about the y-axis?
Answer: A
V = π ∫₁³ x² dy. Since x = y², x² = (y²)² = y⁴. V = π ∫₁³ y⁴ dy.
8. Find the volume generated when the line y = 2x from x = 0 to x = 3 is revolved 360° about the x-axis.
Answer: A
V = π ∫₀³ y² dx = π ∫₀³ (2x)² dx = π ∫₀³ 4x² dx = π [4x³/3]₀³ = π [4273] = 36π units³.
9. Evaluate ∫₀¹ (3x + 2)⁵ dx.
Answer: B
∫₀¹ (3x + 2)⁵ dx = [(3x + 2)⁶ / 18]₀¹ = (5⁶ / 18) - (2⁶ / 18) = 15625 - 6418 = 1556118... Wait: 1556118 = 864.5; let's reduce: 1556118 = 51876 = 17292. Correct option: 1556118 = 309318 is wrong option calculation, 1556118 = 1556118.
10. Evaluate ∫ (2x + 1)³ dx.
Answer: B
Using ∫ (ax + b)ⁿ dx = (ax + b)ⁿ⁺¹ / [a(n + 1)] + c: ∫ (2x + 1)³ dx = (2x + 1)⁴ / [2(4)] + c = (2x + 1)⁴ / 8 + c.