Unit 5: Calculus

Topic 5.11: Definite Integration Questions

Practice 20 exam-style questions for IB Math AA SL Topic 5.11. Review the question stems below, then unlock the full Question Bank to access markschemes, model answers, and AI grading.

1Show that1 mark

2026

Show that the area between them is

a \int b (f (x) - g (x)) d x

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2Show that3 marks

2022

The mean value of a function

f

[a, b]

\frac{1}{b - a} a \int b f (x) d x

. Show that the mean value of

f (x) = x^{2}

[0, 3]

3

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3Find2 marks

2026

Given ∫₀⁶ h(x) dx = 15, find ∫₆⁰ h(x) dx and ∫₆⁶ h(x) dx.

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4Show that1 mark

2026

Show that

2 \int 2 g (x) d x = 0

for any integrable

g

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5Write down2 marks

2026

Write down the value of

1 \int 4 (f (x) - 3 g (x)) d x

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6Find2 marks

2026

Given

0 \int 6 h (x) d x = 15

, find

6 \int 0 h (x) d x

and

6 \int 6 h (x) d x

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7Choose1 mark

Given that

2 \int 9 g (x) d x = 24

, write down

9 \int 2 g (x) d x

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8Evaluate2 marks

2026

Evaluate ∫₀¹ e^3x dx, giving an exact answer.

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9Find3 marks

2026

Find the area between the curve y = x² − 9 and the x-axis from x = 0 to x = 3.

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10Find3 marks

2026

Find the area enclosed between y = 2x and y = x².

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11Find4 marks

2022

The length of the curve

y = x^{2}

from

x = 0

x = 2

is given by

L = 0 \int 2 1 + (\frac{d y}{d x})^{2} d x

. Find

L

, giving your answer to 4 significant figures.

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12Find2 marks

2026

Given ∫₁⁴ f(x) dx = 10 and ∫₁⁷ f(x) dx = 26, find ∫₄⁷ f(x) dx.

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13Evaluate2 marks

2026

Evaluate

0 \int π /2 sin x d x

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14Evaluate2 marks

2026

Evaluate

0 \int 1 e^{3 x} d x

, giving an exact answer.

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15Find3 marks

2026

The curve

y = x^{2} - 2 x

meets the x-axis at

x = 0

and

x = 2

, dipping below it. Find the area between the curve and the x-axis on [0, 2].

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16Find3 marks

2026

Water flows into a tank at a rate

r (t) = 20 - 2 t

litres per minute. Find the volume added between

t = 0

and

t = 5

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17Evaluate3 marks

2026

Evaluate ∫₁² (6x² − 4x + 1) dx.

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18Find3 marks

2026

Find the area enclosed between

y = 4 x - x^{2}

and the x-axis.

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19Evaluate2 marks

2026

Evaluate ∫₀^π/2 sin x dx.

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20Choose1 mark

A function has

2 \int 6 f (x) d x = 21

and

2 \int 11 f (x) d x = 50

. Find

6 \int 11 f (x) d x

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