Definite integrals, areas and the trapezium rule

What you’ll learn

• Evaluate definite integrals and areas between curves.
• Approximate an integral with equally spaced trapezia.

Understand the idea

A definite integral is the change in an antiderivative between two limits. Geometrically, it is signed area: regions below the horizontal axis contribute negatively. For total geometric area, split at crossings and make each region positive. Between two curves, integrate the upper function minus the lower function.

The trapezium rule joins sampled points by straight lines and adds the resulting trapezia. With equally spaced samples, the first and last heights occur once and all interior heights twice. More strips usually improve an estimate for a smooth function. A consistently upward-curving graph lies below its chords, so trapezia overestimate its integral.

Key formulae

$$\int_a^b f(x)\,dx=F(b)-F(a)\quad(F'=f)$$

$$A=\int_a^b[\text{upper curve}-\text{lower curve}]\,dx$$

$$\int_a^b y\,dx\approx\frac h2\left[y_0+y_n+2\sum_{i=1}^{n-1}y_i\right],\qquad h=\frac{b-a}{n}$$

Worked example

Find the area enclosed by $y=5x-x^2$ and $y=x$.

1. Intersections satisfy $5x-x^2=x$, so $x(4-x)=0$ and the limits are $0,4$.
2. The parabola is above the line between them, so integrate $4x-x^2$.
3. $A=[2x^2-x^3/3]_0^4=32-64/3$.

Answer: $A=32/3$ square units.