Use the trapezoidal rule to approximate definite integrals and areas under curves.

How to approximate a definite integral with the trapezoidal rule, set up the calculation from a formula or a table of data, and judge whether the estimate is too high or too low for TCE Mathematics Methods Unit 4.

Generated by Claude Opus 4.78 min answerUpdated 2026-05-30

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What this dot point is asking

Sometimes you cannot integrate a function exactly, or you are only handed measurements at intervals rather than a formula. The trapezoidal rule gives a numerical estimate of the area under the curve in these situations.

The idea

Divide the interval from $x=a$ to $x=b$ into $n$ strips of equal width $h = \dfrac{b-a}{n}$ . Instead of finding the exact area of each strip, approximate it by a trapezium whose parallel sides are the two ordinates (function values) at the edges of the strip. Adding the areas of all the trapezia gives the estimate.

The pattern is easy to remember: the first and last ordinates count once, and every interior ordinate counts twice, all multiplied by half the strip width.

Worked example

Estimating an integral with four strips

Problem

Use the trapezoidal rule with four strips to estimate $\displaystyle\int_0^2 \sqrt{1+x^2}\,dx$ .

Solution

With $a=0$ , $b=2$ and $n=4$ , the strip width is $h = \dfrac{2-0}{4} = 0.5$ . The ordinates are $f(x) = \sqrt{1+x^2}$ at $x = 0, 0.5, 1, 1.5, 2$ :

$y_0 = \sqrt{1} = 1$ , $y_1 = \sqrt{1.25} \approx 1.1180$ , $y_2 = \sqrt{2} \approx 1.4142$ , $y_3 = \sqrt{3.25} \approx 1.8028$ , $y_4 = \sqrt{5} \approx 2.2361$ .

Apply the rule:

\int_0^2 \sqrt{1+x^2}\,dx \approx \frac{0.5}{2}\big[1 + 2.2361 + 2(1.1180 + 1.4142 + 1.8028)\big].

The interior sum is $1.1180 + 1.4142 + 1.8028 = 4.3350$ , doubled to $8.6700$ . Adding the end ordinates gives $1 + 2.2361 + 8.6700 = 11.9061$ , and multiplying by $0.25$ gives

\int_0^2 \sqrt{1+x^2}\,dx \approx 2.977.

Is the estimate too high or too low?

Whether the trapezoidal rule over or underestimates depends on concavity. For a concave up curve the straight tops of the trapezia sit above the curve, so the rule overestimates the true area. For a concave down curve the trapezia tops sit below the curve, so the rule underestimates. Increasing the number of strips reduces the error in either case.

Summary

Choose your strip width $h = (b-a)/n$ , list the $n+1$ ordinates, then combine them as $\dfrac{h}{2}$ times the first plus the last plus twice everything in between. The rule shines when no antiderivative exists or when only tabulated data is available. Use concavity to predict whether the estimate is high or low, and add more strips for a sharper result.