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The Difference Quotient and Integral Calculus

by Vodkacannon
Tags: calculus, difference, integral, quotient
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Vodkacannon
#1
Nov14-12, 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?
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micromass
#2
Nov14-12, 08:39 PM
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P: 18,019
Quote Quote by Vodkacannon View Post
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?
Sure there is a formula for the integral of a function.

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{b-a}{n}\right)\frac{b-a}{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.
Vodkacannon
#3
Nov14-12, 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.

Mark44
#4
Nov15-12, 12:31 AM
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P: 21,216
The Difference Quotient and Integral Calculus

Quote Quote by Vodkacannon View Post
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.
This is in part what the Fundamental Theorem of Calculus says, that differentiation and integration are essentially inverse processes (not inverse functions - these processes operate on functions). One part of this theorem says
$$ \frac{d}{dx} \int_a^x f(t)~dt = f(x)$$
There's some fine print about the continuity of f and such, but the punch line is as above.


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