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The definite integral

Evaluate definite integrals using the fundamental theorem of calculus; interpret the definite integral as the signed area under a curve; work with regions above and below the xx-axis.

Worked examples

Evaluating a definite integral

Straightforward

Problem

Evaluate ∫13(x2−2x+4) dx\displaystyle\int_1^3 (x^2 - 2x + 4)\,dx.

Signed area and regions below the axis

Moderate

Problem

Evaluate ∫02πsin⁡x dx\displaystyle\int_0^{2\pi} \sin x\,dx and explain the result geometrically.

Applying the Fundamental Theorem of Calculus

Challenging

Problem

Define G(x)=∫2x1t2+1 dtG(x) = \displaystyle\int_2^x \frac{1}{t^2 + 1}\,dt. Find G′(x)G'(x) and hence find G′(2)G'(2).

Practise

Q1·Straightforward
Evaluate ∫023x2 dx\displaystyle\int_0^2 3x^2\,dx.
Q2·Straightforward
Evaluate ∫141x dx\displaystyle\int_1^4 \frac{1}{x}\,dx, correct to two decimal places.
Q3·Straightforward
Evaluate ∫0π/2cos⁡x dx\displaystyle\int_0^{\pi/2} \cos x\,dx.
Q4·Straightforward
Evaluate ∫−12(2x+1) dx\displaystyle\int_{-1}^{2} (2x + 1)\,dx.
Q5·Straightforward
Evaluate ∫01ex dx\displaystyle\int_0^1 e^x\,dx, correct to two decimal places.
Q6·Moderate
Evaluate ∫01e2x dx\displaystyle\int_0^1 e^{2x}\,dx, correct to two decimal places.
Q7·Moderate
Evaluate ∫0π/3sin⁡x dx\displaystyle\int_0^{\pi/3} \sin x\,dx. Give your answer as a fraction.
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Q8·Moderate
Evaluate ∫01(x3+2x) dx\displaystyle\int_0^1 (x^3 + 2x)\,dx. Give your answer as a fraction.
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Q9·Moderate
The region between y=x2y = x^2 and the xx-axis for −1≤x≤1-1 \leq x \leq 1 lies entirely above the xx-axis. Find its area. Give your answer as a fraction.
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Q10·Moderate
Evaluate ∫−23(x2−4) dx\displaystyle\int_{-2}^{3} (x^2 - 4)\,dx.
Q11·Challenging
Evaluate ∫1e2x2+1x dx\displaystyle\int_1^e \frac{2x^2 + 1}{x}\,dx, correct to two decimal places.
Q12·Challenging
Define F(x)=∫0x(t2+1) dtF(x) = \displaystyle\int_0^x (t^2 + 1)\,dt. Find F′(3)F'(3).
Q13·Challenging
Find the total area enclosed between the curve y=x3−xy = x^3 - x and the xx-axis for −1≤x≤1-1 \leq x \leq 1. Give your answer as a fraction.

*Note: the curve crosses the xx-axis within this interval, so you will need to handle each sub-region separately.*
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