Approximating an Area with Rectangles

How do you measure the area under the graph of a function $f$ between $a$ and $b$, when the region has no known geometric shape? Riemann's idea is disarmingly simple: cut it up and approximate it with rectangles.

We split the interval $[a \,;\, b]$ into $n$ pieces of equal width. Each piece measures:

$$\Delta x = \frac{b - a}{n}$$

On each piece, we build a rectangle whose height is the value of $f$ at a point of the piece. The sum of the areas of these rectangles approximates the area we are after.

Each rectangle has area height $\times$ width, that is $f(x_i)\,\Delta x$. Adding them all up gives an approximate value of the area:

$$S = f(x_1)\,\Delta x + f(x_2)\,\Delta x + \cdots + f(x_n)\,\Delta x$$

With few rectangles, the approximation is rough: the rectangles overshoot or leave gaps. But the intuition is clear — the thinner the rectangles, that is the larger $n$, the better the approximation. This passage to the limit is what will define the integral, the subject of the next lesson.