# Why do we need to approximate integrals when we can work them out by hand?

Oct 17, 2014

In my opinion, you do not need to unless it takes too long to work out integrals. If integrals are time-consuming, and you do not need an exact value for your purposes, it makes sense to approximate them.

I hope that this was helpful.

May 15, 2015

Not every function (and not every interesting and important function) has an antiderivative that is finitely expressible using the algebraic operations: addition, subtraction, multiplication, division and extraction of roots.

Two examples:

Natural Logarithm
${\int}_{1}^{b} \frac{1}{x} \mathrm{dx}$

The natural log must be approximated using some approximation technique -- by approximating the integral or by some series approximation.

Probablity and Statistics
The standard Normal (or Bell or Gaussian) curve

graph{e^(-1/2 x^2)/sqrt(2pi) [-2.398, 2.469, -0.55, 1.883]}

$\frac{1}{\sqrt{2 \pi}} {\int}_{0}^{z} {e}^{- \frac{1}{2} {x}^{2}} \mathrm{dx}$ gives the probability of a random variable having a standard normal value between $0$ and $z$

This integral cannot be expressed finitely using algebraic operations and must be approximated numerically. (As must ${e}^{x}$ itself.)