Trying to understand why integration is inverse of differentiation

[Taturana]

Greetings,

We know that the two operations (integration and differentiation) are inverse. I mean, if you integrate a function and differentiate the result of the first operation you go back to the first function. But I'm trying to understand WHY. Of course there are a lot of mathematical proofs for this, but I want to understand it intuitively, so I'm trying to understand it geometrically.

I had some progress with the following thought. But it doesn't prove my main question.

Consider the following:

http://img87.imageshack.us/img87/6942/integ.jpg

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I have a random function at the left. I want to calculate the area under its graph from, say, x = 0 to x = 2, so I need to integrate. Geometrically I can divide the area under the graph into small columns (like the green one). Now, in order to calculate the total area under the graph I need to calculate all these columns areas and sum them.

The columns are going to be infinitely small, so if I zoom I will have something like a triangle in the top of the column and a big rectangle from the bottom of the triangle to the x axis. Now to calculate the area of this column, you calculate the area of the rectangle, the area of the triangle and sum them.

Here the differentiation comes in. In order to calculate the area of the triangle, I need to have the value of it's height. And the height of the triangle can be calculated using the tangent of angle theta (the derivative of the function at that x point). Now you do math, take the limits etc etc.

Okay, I already proved that the integration and the differentiation have an intrinsic relation: when I'm integrating (i.e. calculating the area under the curve) I'm constantly getting the derivative at every x's axis point inside the area.

But it doesn't prove that the derivative is the inverse operation of the integration (the antidifferentiation I mean).

Does anyone have any idea of how to prove that (geometrically)?

Thank you for the help.

 

[shaiguy6]

Think of this integral, with respect to your image:

[tex]I(a)=\int_0^a f(x) dx[/tex] a is your endpoint, and obviously the integral changes as you change a. Now, we want to understand why the derivative of I gives back f(x). Basically, think of a changing from some value b to b+epsilon. The amount that a changes between these two points is effectively the derivative. Now, it should be quite clear that by performing this kind of operation step by epsilon step you will begin to outline the shape of the function. Does this help?

 

[lavinia]

Your green region is just the increment in the area from x to x + delta x. the Newton quotient is this area increment divided by delta x and is thus the average height of the function in the green region. As delta x goes to zero this average becomes the value of the function at the left end point.

 

[noone123]

I honestly don't see how this knowledge(or more like understanding) is going to benefit you.

From all I know...Integration is the process of finding areas, volumes etc under curves. The fundamental theorem of calculus is able to use the primitive of the curve to find the area under the curve. So technically, you should be saying that diffentiating and finding the primitive are inverse operations.

lol sorry if I didnt help. I'm quite weak in everything here compared to everyone else

 

[Taturana]

Thank you all for the replies.

Think of this integral, with respect to your image:

[tex]I(a)=\int_0^a f(x) dx[/tex] a is your endpoint, and obviously the integral changes as you change a. Now, we want to understand why the derivative of I gives back f(x). Basically, think of a changing from some value b to b+epsilon. The amount that a changes between these two points is effectively the derivative. Now, it should be quite clear that by performing this kind of operation step by epsilon step you will begin to outline the shape of the function. Does this help?

Yes, I understand, and it does help a little bit. But, generally, you only said what I said in my post. What you said proves that there is a intrinsic relation between the integral and the derivative. But it does not prove that the derivative is the inverse operation of the integration (i.e. antidifferentiation).

Your green region is just the increment in the area from x to x + delta x. the Newton quotient is this area increment divided by delta x and is thus the average height of the function in the green region. As delta x goes to zero this average becomes the value of the function at the left end point.

I don't see how this helps in my question. But thank you anyway!

I honestly don't see how this knowledge(or more like understanding) is going to benefit you.

From all I know...Integration is the process of finding areas, volumes etc under curves. The fundamental theorem of calculus is able to use the primitive of the curve to find the area under the curve. So technically, you should be saying that diffentiating and finding the primitive are inverse operations.

lol sorry if I didnt help. I'm quite weak in everything here compared to everyone else

I know that maybe there is no sense in trying to understand mathematical things intuitively, but I usually understand these things this way.

Like you said: the derivative is the inverse of antidifferentiation (that gives you the primitive) and not of definite integrals (that are used to calculate the areas). I'm afraid that the antidifferentiation itself doesn't represent anything intuitively.

So I'm afraid that after understanding this my question has no sense.

Thank you

 

[lavinia]

I don't see how this helps in my question. But thank you anyway!

Thank you

The intuition is that the area of the green region is the area of a rectangle whose height is the average height of the function. One imagines this height converging to the function's value at the end point as the increment goes to zero. This is a clear picture.

 

[Taturana]

The intuition is that the area of the green region is the area of a rectangle whose height is the average height of the function. One imagines this height converging to the function's value at the end point as the increment goes to zero. This is a clear picture.

Ahh, I see. But it does not tell me why the derivative is the inverse of antidifferentiation, does it?

Thank you again.

 

[lavinia]

Ahh, I see. But it does not tell me why the derivative is the inverse of antidifferentiation, does it?

Thank you again.

yes it does. The area is the integral, the function is the derivative of the area.