Approximating Integrals

What's the point of approximating ??

I never understood this either. Why not just skip all the reimann sums and other shit so professors can spend more time on more difficult integration techniques?

Because most integrals aren't soluble, so you either compute them numerically or solve them in a particular regime.

Go ahead and find me the exact value of the integral of sin(x)/x from 0 to Pi.

>tfw to intelligent for symbolic integration

How do you think a calculator works?

helps you to understand parts of a function without necessarily needing to solve a complex equation or integral

Engineering.

underrated

It takes 0.01% as long and is 99.99% as accurate.

it's exactly [math]Si(\pi)[/math]
where's your god now, piggot?

The exact value of [math] \int_{0}^{x} f(s) \mathrm{d}s [/math] is [math] F(x) [/math], where i define [math] F(x) := \int_{0}^{x} f(s) \mathrm{d}s [/math]. Do I win math now?

Now you're getting it.

it's really the same idea as when we say that [math]\int_1^2 \frac{1}{x} dx = \ln 2 [/math]

>that's the fucking definition of natural log
>what good is knowing that the exact value of the integral is ln2 if we don't know the exact value of ln2

kinda silly since the actual definition of natural log is the inverse function of e^x, so you can find values of log by simply inverting the coordinates of e^x.

Well for one given some [math] f(x) [/math] at random:

$\displaystyle{\int f(x) dx}

probably won't have a solution expressible in elementary terms. Also, outside the ivory tower people just want muh results and most algorithms for numerical integration are pretty efficient.

This

Some people go that route, sure.
But then how do you define e? Typically if you're going that route, e is essentially defined to be that unique number a such that the derivative of a^x is a^x, which is really just a roundabout way of saying that the derivative of the inverse function is 1/x. So it's all the same.

Because Im not an autist that wastes his time looking for analytic solutions in an era where computional power is dirt cheap

>his solutions aren't exact

laughinggirls.jpg

Why would you need exact solutions to a problem you're already approximating?

Because most Integrals have no analytical solution and the only way to solve them is by computation.

There is a whole field of mathematics about that, called numerics.

In the real world calculating integrals is a completely meaningless skill, Integrals as they appear in the real world are almost always solved by a computer.