Integral

The integral is a central concept in calculus. Intuitively, we can think of an integral as a measure of the size of an object with an extent in space. For example, integral calculus lets us calculate the length of a curve, the area of a surface, or the volume of a solid object. The process of calculating integrals is called integration.

A geometric definition

The easiest way to understand integrals is perhaps as a means to calculate area. What do we mean by area in the first place? We do know the precise meaning of area in the case of one simple figure: the rectangle. A rectangle that is $w$ units wide and $h$ units high has area $\scriptstyle w\times h$; let us take this as the definition of area. We can now measure the area of a more complicated shape, such as an apartment floor, by covering it with rectangles, and taking the sum of their individual areas. This is the basic meaning of integration: an integral is simply a sum of smaller parts that together add up to the whole.

Walls are typically at right angles, so tiling a floor with rectangles is no problem. But there are infinitely many kinds of shapes that cannot be exactly covered with rectangles, such as circles, ellipses, or the interior of any curved shape we can draw. Nevertheless, we think of these shapes as having area. We can approximately measure the area of such a shape by covering it with many small rectangles. The more and smaller rectangles we choose, the better the approximation becomes. Using the concept of a limit from mathematical analysis, we can continue to shrink the rectangles until they become infinitely small and the error becomes zero. This process of taking limits is what distinguishes integrals from ordinary sums, and it allows us to exactly calculate lengths, areas, volumes — and so on, of arbitrarily complicated shapes, provided of course that we can express those shapes with exact mathematical formulas.

Let us now give a more formal definition of integral, and also introduce the mathematical notation. Consider a region in the $x$-$y$-plane delimited by the $x$-axis, two vertical lines at $x=a$ and $x=b$, and a curve described by the function $y=f(x)$ as $x$ ranges from $a$ to $b$.

Left: a region bounded by three straight lines and the graph of a function $f$. Right: approximation of the area by rectangles.

We can approximate the area of this region by drawing $n$ rectangles of equal base width along the x-axis, and taking the height of each rectangle to be the height to the function graph anywhere along the extent of the rectangle's base — for example, the rightmost point. Then the $k$'th rectangle from the left has width $(b-a)/n$ and height $h_{k}=f(a+(b-a)(k/n))$ and the sum of all rectangle areas is

$s_{n}={\frac {b-a}{n}}\left(h_{1}+h_{2}+\cdots +h_{n}\right).$

The exact area, $s$, is given by the limit of this expression as $n$ goes to infinity,

$s=\lim _{n\to \infty }s_{n}.$

This limit is called an integral, or more technically, a Riemann integral. Its notation is the following:

$s=\int _{a}^{b}f(x)\,dx$

The equation is pronounced "$s$ equals the integral of $f$ from $a$ to $b$". It is no coincidence that the integral sign, $\scriptstyle \int$, resembles an "S" — it was originally an "S" standing for "sum", but the symbol has changed over time.