What is the connection between differentiation and integration?

[unseensoul]

To find the area under a curve within some range we can either add up infinite infinitesimal tiny rectangles (using limits) (method 1) or do the opposite process of differentiation (method 2).

How did Leibniz/Newton figure out/prove that the opposite process of differentiation (method 2) would give the same answer as adding up rectangles (method 1)?

 

[HallsofIvy]

I honestly can't say precisely how Leibniz or Newton did that, but the proof of the "Fundamental Theorem of Calculus" is given in any calculus book.

Roughly speaking:
First, you show that the "Riemann sum method" (which was developed long after Newton or Leibniz) does in fact give the "area under the curve". You can do that by choosing the height of the rectangles over any interval to be the largest value of the function on that interval so that the sum of the areas of the rectangles, Iⁿn is clearly greater than the area under the curve. Then you turn around and choose the height of the rectangles to be the smallest value of the function on that interval so that the sum of the areas of the rectangles, I_n, is clearly less than the area under the curve. A function is said to be "Riemann Integrable" if, in the limit as the number of intervals goes to 0, Iⁿ and I_n both have the same limit. Since one is always larger than the area and the other is always lower, the common limit must be the area.

Now, given a value x₀, consider the area under the curve, between x₀ and x₀+ h. you can show that area is the difference between the integrals, I(x]sub]0[/sub] and I(x₀+h). Further, using the mean value theorem, you can show that is the area of a rectangle of of base h and height f(x*) where x* is some (unknown) value between x₀ and x₀+ h. That is, I(x₀+ h)- I(x₀)= hf(x*) so (I(x₀+h)- I(x₀))/h= f(x*). Finally, take the limit on both sides as h goes to 0. On the left we get I'(x) and on the right, since x* is always between x₀ and x₀+ h, and h is going to 0, we have f(x). That is I'(x)= f(x)- the derivative of the integral is the original function.

 

[Pere Callahan]

I... A function is said to be "differentiable" if, in the limit as the number of intervals goes to 0, Iⁿ and I_n both have the same limit...

I assume you meant to write Riemann-integrable..?

 

[HallsofIvy]

I assume you meant to write Riemann-integrable..?

Yes, thanks. I will edit my previous response.