Area between curves

Warm-up

Three quick questions from earlier lessons.

Discover

The violet curve is $y = x^2$ , the blue line $y = kx$ . They enclose a region between $x = 0$ and $x = k$ . Move $k$ .

The line y = kx and the parabola y = x². They meet at x = 0 and x = k; between them the line is on top.

On $[0, k]$ , which curve is on top?

The line $y = kx$
The parabola $y = x^2$

Key idea

Area between curves: integrate top minus bottom.

If

f(x) \ge g(x)

on

[a, b]

, the area between them is

\displaystyle\int_a^b \big(f(x) - g(x)\big)\,dx

. The limits are usually where the curves meet; if they cross inside, split there.

Worked example

Area between $y = 2x$ and $y = x^2$ .

Limits

0

and

2

2x = x² at 0 and 2.

\displaystyle\int_0^2 (2x - x^2)\,dx

Top minus bottom.

= 4 - \tfrac{8}{3} = \tfrac{4}{3}

x² − x³/3 from 0 to 2.

Your turn, guided

Your turn: area between $y = 3x$ and $y = x^2$ .

\tfrac{9}{2}

3³/6.

Practice ★

Find the area of the region between $y = 5x$ and $y = x^2$ .
Find the area of the region between $y = 4x$ and $y = x^2$ .
Find the area of the region between $y = 1x$ and $y = x^2$ .

Watch out!

Practice ★★

Find the area of the region between $y = x^2$ and $y = 4x - x^2$ .
Find the area of the region between $y = x^2$ and $y = 2x - x^2$ .

Find the area between $y = 1\cos x$ and $y = 1\sin x$ for $0 \le x \le \dfrac{\pi}{4}$ . Give an exact value.

Exit ticket

Three questions to check you have got it.

Find the area of the region between $y = 5x$ and $y = x^2$ .
Find the area of the region between $y = 4x$ and $y = x^2$ .
Find the area of the region between $y = 1x$ and $y = x^2$ .

Summary card

Area between curves

Find where they meet.
Decide which is on top.
$\int (\text{top} - \text{bottom})\,dx$ .

Watch out: a negative result means the order is reversed.

About this lesson