Talk:Calculus/Definite integral

Archive
I archived all the topics on this page. Some of the topics may still be relevant, but it is difficult to be sure since the content and organization of this page and the whole book have changed significantly since many of those topics were written. If you are looking for suggestions to improve this page, you may want to have a look in the archive. --Greenbreen (discuss • contribs) 17:55, 3 June 2011 (UTC)

Removed material
I removed the following material from this page:

A geometrical proof that anti-derivative gives the area
Suppose we have a function F(x) which returns the area between x and some unknown point u. We don't even know if something like F exists or not, but we're going to assume it does and investigate the implications.

We can use F to calculate the area between a and b, for instance, which is obviously F(b)-F(a). F is something general. Now, consider a rather peculiar situation, the area bounded at $$x$$ and $$x+\Delta x$$ (see Figure 5), in the limit of $$\Delta x \to 0$$. Of course it can be calculated by using F, but we're looking for another solution this time. As the right border approaches the left one, the shape seems to be an infinitesimal rectangle, with the height of f(x) and width of $$\Delta x$$. So, the area reads:


 * $$\mbox{Infinitesimal area} = \lim_{\Delta x \to 0} f(x) \Delta x$$

Of course, we could use F to calculate this area as well:


 * $$\mbox{Infinitesimal area} = \lim_{\Delta x \to 0} F(x + \Delta x) - F(x)$$

By combining these equations, we have


 * $$\lim_{\Delta x \to 0} F(x + \Delta x) - F(x) = \lim_{\Delta x \to 0} f(x) \Delta x$$

If we divide both sides by $$\Delta x$$, we get


 * $$\lim_{\Delta x \to 0} \frac{F(x + \Delta x) - F(x)}{\Delta x} = f(x)$$

which is an interesting result, because the left-hand side is the derivative of F with respect to x. This remarkable result doesn't tell us what F itself is, however it tells us what the derivative of F is, and it is $$f$$. This material was originally on this page, and this page was originally after Indefinite Integrals and the Fundamental Theorem of Calculus. After reordering, I wanted to remove mention of antiderivatives until they were introduced in the Fundamental Theorem of Calculus. I planned to add it to one of those two later pages. However, the argument is basically just the same proof given at the Fundamental Theorem of Calculus, just not as rigorous. So I decided to leave it out altogether. --Greenbreen (discuss • contribs) 18:46, 3 June 2011 (UTC)