hi MathsIsFun

A good start.  On the linked 'Integration Rules' page, I assume the integration by parts and substitution are still in preparation.

Because the fundamental theorem uses areas and some teachers gloss over 'reverse of differentiation' bit, I have met students who have two misconceptions about integration:

(i)  They may think that integration is defined as the reverse of differentiation;

(ii) they may think you can only use it to find areas.

I like your approach as it makes it clear that (i) is not the case.

To avoid (ii) I wonder if you would consider re-wording this bit:

Integration is a method to find the area underneath the graph of a function like this:

Replace with eg. "Integration can be used in many ways such as to find volumes and to calculate, for example, centres of gravity, but it is easiest to start with finding the area underneath the graph of a function like this: "

This has reminded me that I said I would suggest some more for the double differentiation page.  Sorry, this got shunted down my to-do list, but I'll try to get on with this soon.

Bob

Hi Bob, good suggestion.

hi MathsIsFun

Maybe that sentence is a bit long.  So it might be better to split it into several shorter ones.

"Integration is a method for adding up infinesimal 'slices' to calculate a whole. 

It can be used for many things eg. calculating a volume or working out a centre of gravity. 

It is easiest to start with finding the area underneath the graph of a function like this:"

Bob

bob bundy wrote:

hi Stefy

Defining an integral as the area below a graph is equivalent to defining it as the inverse of differentiation and then showing it calculates areas as well.

What!  Why have I spent weeks trying to get 21122012 to do calculus properly.

These statements are not the same.

The fundamental theorem shows that if an area function F(x) can be constructed for the function f(x), then F'(x) = f(x).

So starting with F you can differentiate to get f.

But starting with f you do not necessarily get F.  You might get F + C.

So they are not fully inverses operations to each other.

Bob

They are not inverse in the sense a function is inverse.

The Fundamental theory of Calculus states that the area below the graph of a function is equal to the difference of the antiderivative of the function at the two endpoints of the interval on whic you calculate the area...

The tap/tank example is a great analogy for integration! Big stuff made out of smaller stuff- it's all Calculus!