65 exam-style questions on Definite integrals; areas between curves — topic 5.11 of Calculus in the IB Mathematics: Analysis and Approaches syllabus — with answers and explanations. Samples below; the full set is free with an account.
Calculate ∫ (2x + 1) dx between x = 1 and x = 4.
Answer: 18
An antiderivative is x² + x, so the integral equals (16 + 4) − (1 + 1) = 20 − 2 = 18.
Determine the area of the region enclosed by the line y = x and the curve y = x² between their points of intersection.
Answer: 1/6
The graphs meet at x = 0 and x = 1, with the line above the curve, so A = ∫₀¹ (x − x²) dx = 1/2 − 1/3 = 1/6.
Given that ∫₂⁵ f(x) dx = 7 and ∫₂⁵ g(x) dx = 3, determine the value of ∫₂⁵ (2f(x) − g(x)) dx.
Answer: 11
By linearity of the integral, the value is 2(7) − 3 = 11.
Calculate ∫ sin x dx between x = 0 and x = π.
Answer: 2
An antiderivative of sin x is −cos x, so the integral is −cos π − (−cos 0) = 1 + 1 = 2.
The constant k > 1 satisfies ∫₁ᵏ (1/x) dx = ln 5. Determine the value of k.
Answer: 5
∫₁ᵏ (1/x) dx = ln k − ln 1 = ln k, so ln k = ln 5 gives k = 5.
Determine the area of the region enclosed by the curve y = 4 − x² and the x-axis.
Answer: 32/3
The curve meets the x-axis at x = ±2, so A = ∫₋₂² (4 − x²) dx = (8 − 8/3) − (−8 + 8/3) = 32/3.
That was 6 of 65 Definite integrals; areas between curves questions.
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