Similar to how lim h0 of f(x+h)-f(x)/h is the definition of a derivative, what is the definition of an integral and its explanation? I understand how to get a derivative, but I'm having trouble understanding how a riemann sum becomes an integral.

Question

Similar to how lim h>0 of f(x+h)-f(x)/h is the definition of a derivative, what is the definition of an integral and its explanation? I understand how to get a derivative, but I'm having trouble understanding how a riemann sum becomes an integral.

kainui · Answer

$$dy=\int\limits_{b}^{a}f(x)dx$$Does the sum of the function from a to b multiplied by an infinitely small amount of x equal the infinitely small amount of y? Or the area? I'm kinda confused on how Riemann sums turn into integrals.

shayaan_mustafa · Answer

you know one thing. integral can only compute continuous data. while summation can only compute discrete data.

shayaan_mustafa · Answer

what are asking for actually?

formula in question is known as first principle. and$$\lim_{h \rightarrow 0}$$

turingtest · Answer

$$\int_{a}^{b}f(x)dx=\lim_{n \rightarrow \infty}\sum_{i=1}^{n}f(a+i\Delta x)\Delta x$$where$$\Delta x=\frac{b-a}n$$the interpretation of that would take quite a while to explain here, so let me find a good link...

turingtest · Answer

the definition is written a bit differently here, but it should give the same result http://tutorial.math.lamar.edu/Classes/CalcI/DefnOfDefiniteIntegral.aspx

kainui · Answer

I guess that helps a little. In particular, why does the i=1 at the bottom of the Riemann sum instead of i=dx?

kainui · Answer

I'll clarify a little, doesn't that mean that we're adding by n+i, and if we're adding up everything below the curve, doesn't that just amount to adding up every integer and missing every point in between, so adding up in segments of dx would seem right in my mind, I think I have it wrong though in my mind, I just need to rectify it... Sorry!

turingtest · Answer

i=dx doesn't really make any sense mathematically
dx is a differential quantity that comes from taking the limit
in the Riemann sum you haven't taken the limit yet, so dx can't show up.
Furthermore i is an 'index' to keep count by. dx is a differential. They are totally different things.

Let me try to draw a little picture|dw:1327956162330:dw|In the drawing above you can see that Delta x=(b-a)/n is the width of each rectangle. 

The region starts at point a and moves in increments of Delta x. Each rectangle has height f(a+Delta x).

Now here's the kicker:
Imagine the rectangles getting smaller and smaller. That will approximate the area better and better...