Kainui
  • Kainui
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.
Mathematics
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schrodinger
  • schrodinger
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Kainui
  • Kainui
\[dy=\int\limits_{b}^{a}f(x)dx\]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
  • Shayaan_Mustafa
you know one thing. integral can only compute continuous data. while summation can only compute discrete data.
Shayaan_Mustafa
  • Shayaan_Mustafa
what are asking for actually? formula in question is known as first principle. and\[\lim_{h \rightarrow 0}\]

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TuringTest
  • TuringTest
\[\int_{a}^{b}f(x)dx=\lim_{n \rightarrow \infty}\sum_{i=1}^{n}f(a+i\Delta x)\Delta x\]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
  • TuringTest
the definition is written a bit differently here, but it should give the same resulthttp://tutorial.math.lamar.edu/Classes/CalcI/DefnOfDefiniteIntegral.aspx
Kainui
  • Kainui
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
  • Kainui
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
  • TuringTest
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...
TuringTest
  • TuringTest
Rectangles getting smaller means more of them, so n (the number of rectangles) is getting larger. The width of each rectangle gets smaller, which means Delta x gets infinetesimally small (so yes, it approaches dx), however we can't use dx in the formula because then we don't have n and the whole thing will blow up to infinity. dx only appears after we take the limit. Hence as n goes to infinity the area is exact! hence the integral is the limit of the Riemann sum as the number of rectangles approaches infinity.
TuringTest
  • TuringTest
\[\lim_{n \rightarrow \infty}\Delta x=dx\]so that's the last step basically
Kainui
  • Kainui
I guess the height can also be something a little different like the average of a and b to be a riemann sum centered in the middle of the rectangle? I think this is starting to fit with what I know now. I need to think about it for a little while and work it out on my own. I keep seeing this f(xi*) notation though, and haven't seen any obvious reference as to why there is an astrisk there. Thanks for your help so far.
TuringTest
  • TuringTest
|dw:1327956923592:dw|you get the idea hopefully