Area between curves

The idea

The area between two curves is $\int_a^b (\text{top} - \text{bottom})\,dx$ , where $a$ and $b$ are where they meet. For a parametric curve, the area under it is $\int y\frac{dx}{dt}\,dt$ with $t$ -limits.

Solve top $=$ bottom to find the limits.
Integrate top minus bottom between them.
Parametric: replace $dx$ with $\frac{dx}{dt}\,dt$ and use $t$ limits.

Watch out If the curves cross inside the interval, split the area at the crossing.

See it worked

Worked example

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

They meet at $x = \pm 2$ .
$\int_{-2}^{2} (4 - x^2)\,dx = \left[4x - \frac{x^3}{3}\right]_{-2}^{2}$ .
$= \frac{16}{3} - \left(-\frac{16}{3}\right) = \frac{32}{3}$ .

Answer $\frac{32}{3}$

Your turn

Question 1Non-calculator3 marks

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

Question 2Non-calculator3 marks

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

Question 3Non-calculator3 marks

A curve has $x = t^2$ , $y = 2t$ . Find the exact area between the curve, the $x$ -axis and $x = 4$ , for $t$ from $0$ to $2$ .

In past papers

19 official questions in 18 papers since 2018. The latest:

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