Approximating Integrals

>>8911833
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?

>>8911833
Because most integrals aren't soluble, so you either compute them numerically or solve them in a particular regime.

>>8911852
Go ahead and find me the exact value of the integral of sin(x)/x from 0 to Pi.

File: woj_smart_crane.jpg (169KB, 1930x1698px) Image search: [Google]

169KB, 1930x1698px

>tfw to intelligent for symbolic integration

>>8911833
How do you think a calculator works?

>>8911833
helps you to understand parts of a function without necessarily needing to solve a complex equation or integral

>>8911833
Engineering.

>>8911861
underrated

>>8911833
It takes 0.01% as long and is 99.99% as accurate.

>>8911861
it's exactly [math]Si(\pi)[/math]
where's your god now, piggot?

>>8912192
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?

>>8912296
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

>>8912309
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.

>>8911854
>>8912163

This

>>8912452
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.

>>8911833
Because Im not an autist that wastes his time looking for analytic solutions in an era where computional power is dirt cheap

>>8912784
>his solutions aren't exact

laughinggirls.jpg

>>8912853
Why would you need exact solutions to a problem you're already approximating?

>>8911833
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.