Definite integrals and areas

The idea

$\int_a^b f(x)\,dx = F(b) - F(a)$ , where $F$ is the integral of $f$ . It gives the area between the curve and the $x$ -axis, but area below the axis comes out negative.

Integrate, without $+ c$ , and write it in square brackets.
Substitute the top limit, then subtract the bottom limit.
If the curve dips below the axis, find each part separately and make each positive.

Watch out Subtract the whole bottom-limit value, brackets and all.

See it worked

Worked example

Evaluate $\int_1^3 (3x^2 + 2)\,dx$ .

$\left[x^3 + 2x\right]_1^3$ .
$= (27 + 6) - (1 + 2)$ .
$= 30$ .

Answer $30$

Your turn

Question 1Non-calculator3 marks

Evaluate $\int_0^2 (x^3 + 1)\,dx$ .

Question 2Non-calculator3 marks

Find the exact area enclosed by $y = 4x - x^2$ and the $x$ -axis.

Question 3Non-calculator3 marks

The curve $y = x(x - 3)$ lies below the $x$ -axis between $x = 0$ and $x = 3$ . Find the area between the curve and the axis there.

In past papers

48 official questions in 43 papers since 2018. The latest:

All A-level Maths past papers

A-level Maths Pro, Pearson Edexcel 9MA0

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