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The Difference Quotient and Integral Calculus 
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#1
Nov1412, 08:29 PM

P: 39

I'm just a high school senior who noticed that the derivative has a general formula that we all know is,
[itex]\frac{f(x+h)f(x)}{h}[/itex] but that there is no general formula (at least I haven't heard of it yet) for the integral of a function. I know I cannot simply just take the inverse of the difference quotient. Is it impossible to generalize a formula for the integral? 


#2
Nov1412, 08:39 PM

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P: 18,330

For example, let [itex]f:[a,b]\rightarrow \mathbb{R}[/itex], then we can write (if the integral exists) [tex]\int_a^b fdx=\lim_{n\rightarrow +\infty} \left[\sum_{k=1}^n f\left(a+k\frac{ba}{n}\right)\frac{ba}{n}\right][/tex] This is a possible formula for the integral. Of course, there are some issues, for example, I partitioned [a,b] in a certain way and I let f act on the partition in a certain way. It must be clarified that these choice don't matter (and for which functions they don't matter!). But all in all, this formula can be used for most functions. 


#3
Nov1412, 08:44 PM

P: 39

Interesting. I had never seen that before. I can see why you're using a summation operator because integration is just, in geometric terms, summing up infinitsimaly small parts to make a whole.
Thanks. That was a fast reply. So I don't start another thread, may I ask if you were to compose the functions of the integral and derivative of a function, f(g(x)) or g(f(x)), you should receive the origional function right? If this is true then this is one way to proove that integration and derivation are inverses. 


#4
Nov1512, 12:31 AM

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P: 21,307

The Difference Quotient and Integral Calculus
$$ \frac{d}{dx} \int_a^x f(t)~dt = f(x)$$ There's some fine print about the continuity of f and such, but the punch line is as above. 


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