Trapezium rule

#### Tier: #Higher

Description

The trapezium rule is a method for estimating the area under a curve by splitting it into a number of trapezoids (strips).

$$\text{Area} \approx \frac{1}{2}h\left[(y_0 + y_n) + 2(y_1 + y_2 + \cdots + y_{n-1})\right]$$

where $h$ is the width of each strip and $y_0, y_1, \ldots, y_n$ are the heights (function values) at each vertical boundary.

Method:

1. Divide the interval into $n$ equal strips of width $h$
2. Calculate $y$-values at each boundary
3. Apply the formula: add the first and last, then twice all the middle values

Example: Estimate $\int_0^4 x^2,dx$ using 4 strips ($h=1$, values at $x=0,1,2,3,4$: $y=0,1,4,9,16$): $$\approx \frac{1}{2}(1)\left[(0+16)+2(1+4+9)\right] = \frac{1}{2}[16+28] = 22$$

The exact answer is $\frac{64}{3} \approx 21.3$, so the approximation is an overestimate (curve is convex).

Common error: forgetting to double the middle values, or miscounting the strips.

Links

Areas and volumes

Use the TABLE function to generate $y$-values quickly for any function.