Question

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

Answer 1

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.

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I hope that this was helpful.

Answer 2

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 1/x 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]}

`1/sqrt(2pi) int_0^z e^(-1/2 x^2)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.)