Revise The trapezoidal rule and numerical integration for IB Mathematics: Applications and Interpretation (5.4)

Finding an antiderivative is the standard way to evaluate a definite integral, but for many functions no elementary antiderivative exists. Moreover, in science and engineering we often collect data at discrete points without knowing the function that links them — we might record a car's speed every second without ever having an equation for it. In these cases we approximate the value of the integral numerically, and the trapezoidal rule is the tool Applications and Interpretation asks you to use.

Divide the interval $[a,b]$ into $n$ vertical strips of equal width $h$. Over each strip we replace the curve with the straight line joining the two points on it, turning each strip into a trapezoid. A trapezoid with parallel vertical sides $y_0$ and $y_1$ and width $h$ has area:

$A_{\text{single}} = \frac{1}{2}(y_0 + y_1)h$

Adding the areas of all $n$ trapezoids and collecting terms — each interior ordinate is shared by two adjacent trapezoids, so it appears twice — gives the trapezoidal rule in the formula booklet.

$\int_a^b y\,dx \approx \frac{h}{2}\left(y_0 + y_n + 2(y_1 + y_2 + \dots + y_{n-1})\right), \qquad h = \frac{b-a}{n}$

Because each trapezoid replaces the curve with a straight chord, whether the estimate is too big or too small depends only on which side of the chord the curve sits — that is, on the concavity. Where the curve is concave up ($\cup$-shaped, $f''>0$) the chord lies above the curve, so the trapezoids include extra area and the rule OVER-estimates. Where the curve is concave down ($\cap$-shaped, $f''<0$) the chord lies below the curve, so the trapezoids miss some area and the rule UNDER-estimates. If the concavity changes across the interval you cannot decide from concavity alone.

The trapezoidal rule is especially useful for data from experiments or observations, where there is no formula at all. Measure the cross-sectional profile of a river and you can estimate its area to find the flow rate; record the velocity of a runner at intervals and you can estimate the distance travelled as the area under the velocity–time graph.

On Paper 2 the marks are analytic: each is tied to a specific line of working. A method mark (M) rewards a correct approach even if the arithmetic later slips; an accuracy mark (A) rewards a correct result and is usually dependent on the method mark being earned. Follow-through (FT) means a correct final step performed on your own earlier (wrong) value still scores, and equivalent correct forms are accepted. All of that protection exists only if your method is on the page. Study how each mark below is earned by a specific line.

The trapezoidal rule is your fallback whenever an exact integral is out of reach — a genuinely useful skill for modelling with real data, where formulas rarely exist. Keep the habit: find $h$, list the $n+1$ ordinates, double only the interior ones, multiply by $\frac{h}{2}$, and check the concavity to say whether the estimate is high or low. The same area-under-a-curve thinking underpins the definite integrals and modelling problems that follow.