Area under a curve

5 minAS Level1.8.4

AI-drafted, independently checked · not yet teacher-reviewed

By the end of this lesson

By the end you can find the area of a region between a curve, the $x$ -axis and lines like $x = 1$ and $x = 3$ , including regions below the axis.

What they ask

Find the area of the region bounded by the curve, the $x$ -axis and the lines $x = 1$ and $x = 3$ .

Or there's a shaded region on a diagram, and you find the limits yourself from where the curve crosses the $x$ -axis.

Definition

Area under a curve

For a curve above the $x$ -axis from $x = a$ to $x = b$ :

$\text{Area} = \int_a^b y\,dx$

$y$ is the curve's equation, written in terms of $x$ .
$a$ and $b$ are the $x$ -values where the region starts and stops (the limits).

Why an integral gives area

Slice the region into very thin vertical strips. Each strip is almost a rectangle: height $y$ , width a tiny bit of $x$ .

Adding up all the strips gives the area. As the strips get thinner, that sum becomes exactly $\displaystyle\int_a^b y\,dx$ .

The method

Find the limits: the lines they give you, or where the curve meets the $x$ -axis (solve $y = 0$ ).
Integrate $y$ and work out top limit minus bottom limit.
Check which side of the axis the region is on. Below it, the integral comes out negative, and the area is its size.
Region crossing the axis? Split it at the crossing and add the sizes.

Worked example: The easiest one: check it with a trapezium

Find the area between the line $y = 2x + 1$ , the $x$ -axis, and the lines $x = 1$ and $x = 3$ .

Have a go first?

Quick check

CalculateQuestion 1 · Recall

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