4.2 · Riemann sums, signed area, and numerical integration

A sum of rate times interval widths

A Riemann sum adds $\displaystyle f(x_i^*)\Delta x_i$ across subintervals. Each product has units of output rate times input units. Unequal intervals need their own widths; never silently use one common width for uneven data.

As the maximum subinterval width approaches zero, the sums define the definite integral for continuous functions. Positive and negative values contribute with their signs.

$$\int_a^b f(x)\,dx=\lim_{n\to\infty}\sum_{i=1}^{n}f(x_i^*)\Delta x$$

Choose the sampling rule explicitly

A left sum uses the left endpoint of each interval; a right sum uses the right endpoint. A midpoint sum uses the interval midpoint. A trapezoidal approximation uses interval width times the average of its endpoint heights.

For an increasing function, a left sum underestimates and a right sum overestimates the integral. For a concave-up graph, trapezoids overestimate; the midpoint rule underestimates. Describe the required behavior over the whole interval.

Use a graph or a table without inventing data

A piecewise-linear graph can often be integrated exactly with triangles, rectangles, or trapezoids. Regions below the axis subtract from the signed integral but add to geometric area.

Tabular values alone do not determine the exact integral unless extra information is supplied. Report a numerical method as an approximation, and use given midpoint values only when they correspond to the stated subintervals.