Riemann sums

A definite integral is defined as the limit of a sum of the areas of thin rectangles¹. We confirm it for $f(x) = x^2$ from $0$ to $2$ .

The sum built on left endpoints

Cut the interval into eight equal pieces of width $\dfrac{1}{4}$ and stand a rectangle over each, taking the value at the left endpoint as its height.

\frac{1}{4}\sum_{k=0}^{7} \left( \frac{k}{4} \right)^2 = \frac{1}{64} (0 + 1 + 4 + 9 + 16 + 25 + 36 + 49) = \frac{35}{16}

That is $\dfrac{35}{16} = 2.1875$ , smaller than the true value $\dfrac{8}{3} \approx 2.6667$ . Using the height at the left endpoint underestimates an increasing function.

Trapping the value between left and right

Construction	Value	Relation to the true value
Left endpoints	$\dfrac{35}{16} = 2.1875$	smaller
The true value	$\dfrac{8}{3} \approx 2.6667$	the reference
Right endpoints	$\dfrac{51}{16} = 3.1875$	larger

The gap between them is $1$ , which is the difference of the end heights $f(2) - f(0) = 4$ multiplied by the width $\dfrac{1}{4}$ . The true value always lies between the two.

Refining the partition

The left-endpoint sum for $n$ pieces works out as follows.

\begin{align*} \frac{2}{n}\sum_{k=0}^{n-1} \left( \frac{2k}{n} \right)^2 &= \frac{8}{n^3} \cdot \frac{(n-1)n(2n-1)}{6} \\ &\longrightarrow \frac{8}{3} \quad (n \to \infty) \end{align*}

That limit is the value of the definite integral.

The general form

Cutting $[a, b]$ into $n$ equal pieces of width $\Delta x = \dfrac{b - a}{n}$ , the limit of the sum is the definite integral.

\int_a^b f(x) \, dx = \lim_{n \to \infty} \sum_{k=0}^{n-1} f(a + k \Delta x) \, \Delta x

It looks like a computation of area, but it is really a limit of addition. That is why every quantity of the form multiply and add can be written as a definite integral.

Distance from velocity
Mass from density
A total from a unit price

The parabola on the graph is $y = x^2$ , the staircase is the tops of the rectangles of width $\dfrac{1}{4}$ , and the large dots are the two ends of the interval.

Riemann sum, Wikipedia