H2 Maths: Why Rectangles Help Explain a Definite Integral

The rectangles explain the sum

An SGExams student revising H2 Mathematics questioned why a task required rectangles when an integral seemed available. The drawing has a mathematical purpose: it can show what a sum represents, how its terms relate to an interval and why a limit becomes an integral.

The 2026 H2 Mathematics 9758 syllabus includes the definite integral as a limit of a sum. This guide uses an original example to connect those representations. It is not a prediction of an examination question, and the exact task may require more than evaluating an integral.

Identify the width before choosing a height

Consider the invented sum Sₙ = (1/n)Σ(k/n)², where the summation runs from k = 1 to n. The interval is from 0 to 1 and there are n equal subintervals, each of width 1/n. The factor outside the summation supplies that width.

The sample positions are x = k/n, the right-hand endpoints of those subintervals. Each height is therefore f(k/n) for f(x) = x². One term is width multiplied by height. Do not identify a height as an area or forget the width simply because the summation notation looks complicated.

Use a small number of rectangles first

For n = 2, each width is 1/2. The right-endpoint heights are (1/2)² and 1², so S₂ = (1/2)(1/4 + 1) = 5/8. A sketch of y = x² between 0 and 1 shows that these right-endpoint rectangles lie above the curve on their subintervals.

That overestimate depends on the function increasing on this interval and the use of right endpoints. It is not a rule that every rectangle sum overestimates an integral. Changing the endpoint choice or the behaviour of the function can change the comparison.

Connect the limit to the integral

As n increases, the subinterval width tends to zero. For this continuous function, the sums approach the definite integral of x² from 0 to 1, which equals 1/3. The value 5/8 from two rectangles is an approximation, not the limit.

An integral can be signed: regions below the x-axis contribute negatively to the usual definite integral. If a question asks for total geometric area, inspect signs and intervals rather than automatically calling the integral a positive area. Our example stays above the axis, so that distinction causes no complication here.

Translate an unfamiliar sum systematically

Write down the interval, number of subdivisions, width, sample positions and function values. Check whether the endpoints in the sum agree with the diagram. On a general interval from a to b, equal widths are (b − a)/n, not automatically 1/n.

Choose an appropriate limit-of-sum question from our H2 Maths papers. Sketch a small case and label a single rectangle before manipulating the complete sum. If asked to explain an inequality, state the relevant property of the function and endpoint choice. Computing the integral alone may leave the requested explanation unanswered.

Sources and further reading

Checked on 4 November 2026. The linked discussion is real; the examples and practice routines are our original guidance. Older forum posts are not evidence of current examination rules.

• H2 MATH. — 19 August 2026.
• SEAB H2 Mathematics 9758 syllabus, 2026