Why Is the Trapezoidal Rule Useful for Irregular Areas Without a Known Function?

The trapezoidal rule is useful because it estimates an irregular area directly from measured widths or heights, even when no equation models the boundary. It replaces the unknown curved boundary with straight-line segments, creating trapezoids whose areas can be calculated.

How the Method Works

Suppose measurements of height are taken at equal horizontal intervals of width $h$. Between each pair of adjacent measurements, the method assumes the boundary is approximately straight.

The IB formula is

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

The first and last ordinates appear once, while every interior ordinate appears twice because it belongs to two neighbouring trapezoids.

This makes the method especially suitable for river cross-sections, land surveys, and other situations where data are recorded at fixed positions rather than generated by a formula. Each strip has area $\frac{h}{2}(y_i+y_{i+1})$, and adding all strip areas produces the compact formula above.

For example, measurements across an irregular region are taken every $2\text{ m}$:

Using $h=2$,

$$A \approx \frac{2}{2}\left[2+1+2(3.5+4+3)\right]=24\text{ m}^2.$$

Therefore, the irregular area is estimated as $24\text{ m}^2$. This is an approximation, because the actual boundary may curve between measurements. More closely spaced measurements usually improve the estimate because shorter straight segments follow the boundary more accurately.

The estimate may be above or below the true area. Without knowing the boundary’s curvature, do not claim that the trapezoidal rule always overestimates or always underestimates.

A common misconception is that “irregular area” means the intervals may be unequal. In IB Mathematics: Applications and Interpretation SL 5.8, the standard formula assumes equal-width intervals. If the widths differ, one value of $h$ cannot be used for the whole table.

Exam Technique

In a Paper 1 or Paper 2 response, write the trapezoidal-rule expression before the GDC value, double only the interior ordinates, include squared units, and state that the result is approximate. Keep full calculator precision until the final answer.