# What is integration?

Mar 5, 2016

Roughly speaking, integration is the inverse of differentiation, but there are several ways to think about it...

#### Explanation:

Given a suitably well behaved function $f \left(x\right) : \mathbb{R} \to \mathbb{R}$, and an interval $\left(a , b\right)$, the definite integral ${\int}_{a}^{b} f \left(x\right) \mathrm{dx}$ is the "area under the curve" between $a$ and $b$.

At any particular point $t \in \mathbb{R}$, the rate of change of area as you increase $t$ is equal to the $f \left(t\right)$. That is, the derivative of the integral is equal to the original function.

Integration covers a lot more cases than just Real valued functions of Real numbers. You can integrate over any kind of measurable set - e.g. a plane, a curve, a surface, a volume. The function that you are integrating may have any kind of value that is possible to sum and multiply by a scalar, e.g. Real, Complex, vector.

In such contexts you can think of an integral as a sort of infinite sum of values of a function over infinitesimally small pieces of the set over which you are integrating.

For example, suppose you have a function $f \left(p\right)$ defined for points on the surface $S$ of a sphere, with surface area $A$. Then the average value of $f \left(p\right)$ over the surface of the sphere is:

$\frac{{\int}_{p \in S} f \left(p\right) \mathrm{dp}}{A}$

If we split the surface of the sphere into a large number of little patches ${S}_{i}$ of areas ${A}_{i}$, each containing a representative point ${p}_{i} \in {S}_{i}$, then we could approximate the integral over the surface:

${\int}_{p \in S} f \left(p\right) \mathrm{dp} \approx {\sum}_{i} {A}_{i} f \left({p}_{i}\right)$