Why do we need to approximate integrals when we can work them out by hand?
2 Answers
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.
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
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]}
This integral cannot be expressed finitely using algebraic operations and must be approximated numerically. (As must