10 questions · Form 5 Additional Mathematics Bab 3: Integration
Find the area bounded by the curve x = y² - 4 and the y-axis from y = 0 to y = 2.
Prefer reading to quizzing? All 10 questions are listed below with the answer and explanation under each one.
1. Find the area bounded by the curve x = y² - 4 and the y-axis from y = 0 to y = 2.
Answer: A
Area = |∫₀² (y² - 4) dy| = |[y³/3 - 4y]₀²| = |(83 - 8) - 0| = |-163| = 163 units².
2. 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.
3. Find ∫ (6x³ - 2) dx.
Answer: A
∫ (6x⁻³ - 2) dx = 6(x⁻²/-2) - 2x + c = -3x⁻² - 2x + c = -3x² - 2x + c.
4. Given that ∫₁⁴ f(x) dx = 7, evaluate ∫₁⁴ [2f(x) + 3] dx.
Answer: A
∫₁⁴ [2f(x) + 3] dx = 2 ∫₁⁴ f(x) dx + ∫₁⁴ 3 dx = 2(7) + [3x]₁⁴ = 14 + (12 - 3) = 14 + 9 = 23.
5. 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.
6. 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.
7. 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.
8. Evaluate ∫₀⁴ (1 / √(2x + 1)) dx.
Answer: A
∫₀⁴ (2x + 1)⁻¹/² dx = [(2x + 1)¹/² / (2 × 12)]₀⁴ = [√(2x + 1)]₀⁴ = √9 - √1 = 3 - 1 = 2.
9. The area bounded by y = kx² from x = 0 to x = 2 is 8 units². Find the value of k.
Answer: B
∫₀² kx² dx = 8 => [kx³/3]₀² = 8 => 8k3 = 8 => k = 3.
10. Given ∫₁³ f(x) dx = 5, what is the value of ∫₃¹ f(x) dx?
Answer: D
By definite integral property, ∫₃¹ f(x) dx = -∫₁³ f(x) dx = -5.