Use the trapezoidal rule to estimate areas and definite integrals, and determine whether the estimate is an over- or under-estimate

What this dot point is asking

NESA wants you to estimate the area under a curve, or equivalently the value of a definite integral, by slicing the region into vertical strips and treating the top of each strip as a straight line rather than a curve. Each strip becomes a trapezium, you add the trapezium areas, and that total approximates $\int_a^b f(x)\,dx$. You also need to say whether the estimate is too big or too small by looking at the concavity of the curve.

This is the tool for the situation where exact integration is unavailable: either you are only given a table of measured values (a surveyor's offsets, a car's speed at fixed times) and have no formula at all, or you have a formula whose integral is beyond the course. The trapezoidal rule turns either case into a few multiplications and additions.

The answer

The idea is to replace the curved top of each strip with the straight chord joining its two end points. A strip with vertical sides of height $f(a)$ and $f(b)$ and width $b - a$ is then a trapezium, and the area of a trapezium is the average of the two parallel sides times the distance between them.

Why the interior ordinates are doubled

The multi-strip formula is not a new rule, just the single rule applied to each strip and the results added. Take three strips with ordinates $y_0, y_1, y_2, y_3$ and common width $h$. The three trapezia have areas

Add them and factor out $\frac{h}{2}$:

Every interior ordinate is shared by two adjacent trapezia, so it appears twice; the two outermost ordinates belong to only one trapezium each, so they appear once. That is the whole reason for the "ends once, middles twice" pattern, and seeing it this way means you never have to memorise which values get doubled. The single-application rule is just the $n = 1$ case, where there are no interior ordinates at all.

The multi-strip rule, strip by strip

The figure shows four strips ($n = 4$). The five vertical lines are the ordinates $y_0$ through $y_4$, equally spaced $h$ apart along the $x$-axis. Each shaded trapezium has a flat top (the chord), and stacking the four areas with the "ends once, middles twice" weighting gives the estimate. The more strips you use, the more closely the chords hug the curve, so a larger $n$ gives a more accurate estimate. In the exam you use exactly the number of strips the question (or its table) dictates: "two applications of the trapezoidal rule" means $n = 2$, and a table with five columns of values means $n = 4$.

Reading the ordinates: from a table or from a formula

There are two ways the ordinates $y_0, \dots, y_n$ reach you.

From a table. The question hands you a table of $x$-values and matching $f(x)$-values. Read the heights straight off it; you do not need (and may not have) a formula. This is the common case when the data comes from measurement, and it is exactly how the 2024 exam supplied $\ln(1 + x^2)$ to 4 decimal places. Check first that the $x$-values are equally spaced, because the rule in this form requires a constant strip width $h$.

From a formula. The question gives $y = f(x)$, and you choose (or are told) how many strips to use, work out $h = \frac{b - a}{n}$, list the $x$-values $a, a + h, a + 2h, \dots, b$, and substitute each into $f$ to get its ordinate. Set the working out as a small table of $x$ against $y$ so you do not mismatch a height with the wrong strip.

Either way, the arithmetic of the rule is identical once you have the ordinates.

Estimating a definite integral

Because the area under $y = f(x)$ from $a$ to $b$ is the definite integral $\int_a^b f(x)\,dx$ (for $f(x) \ge 0$), the trapezoidal rule is also a way to estimate an integral you cannot evaluate exactly. The phrasing "use the trapezoidal rule to estimate $\int_a^b f(x)\,dx$" is asking for precisely the same calculation as "estimate the area under the curve": list the ordinates, weight the ends once and the middles twice, multiply by $\frac{h}{2}$.

Over-estimate or under-estimate: read the concavity

The trapezoidal rule replaces the curve with straight chords, so the only question is whether each chord sits above or below the curve it is cutting across, and that is decided by concavity.

If the curve is concave up ($f''(x) > 0$, the curve bends upward like a valley) on the interval, every chord lies above the curve, so each trapezium has a little extra area between the chord and the curve. The rule overestimates.

If the curve is concave down ($f''(x) < 0$, the curve bends like a hill) on the interval, every chord lies below the curve, so each trapezium misses a sliver of area under the curve. The rule underestimates.

A clean way to remember which is which: a chord always joins two points on the curve, so it is a straight shortcut between them. On a valley (concave up) the shortcut runs over the dip, so it is too high and the area comes out too big. On a hill (concave down) the shortcut cuts under the bulge, so it is too low and the area comes out too small.

How exam questions ask about the trapezoidal rule

The wording is fairly fixed, and it tells you exactly which form of the rule to use:

• "Use the trapezoidal rule to estimate the area / the shaded region / $\int_a^b f(x)\,dx$." The default request. Read or compute the ordinates and apply $\frac{h}{2}[\,\text{ends} + 2(\text{middles})\,]$.
• "Using the function values provided ..." or a table is given. Read $y_0, \dots, y_n$ straight off the table; the number of columns fixes $n$ (five columns means four strips). Check the $x$-values are evenly spaced before using $h$.
• "Use two applications of the trapezoidal rule" (or one, or $n$ applications). "Applications" counts the strips: two applications means $n = 2$, so $h = \frac{b - a}{2}$ and three ordinates. One application is the single-trapezium rule.
• "Use the trapezoidal rule with $n = 4$ strips" or "... with sub-intervals of width $0.5$." Either form pins down $h$; if width is given, $n = \frac{b - a}{h}$.
• "Is your answer an over-estimate or an under-estimate? Give a reason." Decide from concavity: concave up over, concave down under. If a graph is shown or $f''$ was found earlier in the question, quote that as your reason.
• "Hence" after estimating an area exactly and approximately. The two values are sometimes combined to bound a constant (the 2025 paper used the area to deduce an inequality for $e$), so keep both your exact and estimated answers.

The trap to watch is the meaning of "applications" or "strips" versus "ordinates": $n$ strips always need $n + 1$ ordinates, and the strip width is $h = \frac{b - a}{n}$, not $\frac{b - a}{n+1}$.