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# Riemann sums in summation notation

AP.CALC:
LIM‑5 (EU)
,
LIM‑5.B (LO)
,
LIM‑5.B.2 (EK)
Generalizing the technique of approximating area under a curve with rectangles. Created by Sal Khan.

## Want to join the conversation?

• Just wondering. How is this different from a right Riemann sum or a midpoint Riemann sum?
• I'm no expert in calculus (I'm just learning this now), but I'm guessing that depending on the curve, the three might all be different. If you use f(x) = x^2 as an example, and find the interval from x=2 to x=5, if you draw it on paper, you'll see that a left Riemann sum is an underestimate, a right Riemann sum is an over estimate, and a midpoint Riemann sum is somewhere in between. (But remember this is not always the case.) So depending on the curve, you might want to use a different one.
• where did that "i" come from and what is it's purpose?
i get that the bottom of the Sigma is the starting rectangle and the top is the finish ("N") but then the i re-appears in subtext of x. any help?
• "i" is just a letter for number of rectangles, starting with 1 and ending with n. As for the subtext, it is just pointing out that for rectangle 1 (i=1), x(i-1) (in this case x0) is used and for rectangle 2 (i=2), x(i-1) (in this case x1) is used... and so on until rectangle n (i=n), where x(n-1) is used. Hope that helps.
• I am vaguely familiar with Riemann's work and, from what I understand, wasn't the stuff he was doing much, much more advanced than this? Why would he waste his time doing these sums?
• Although integrals had been studied long before Riemann, it was his work that led to a far more rigorous approach and formal definitions. So, he was advancing integral calculus and making it far more rigorous. So, this was not unimportant work even if it seems too simplistic nowadays.

Though, of course, Riemann was one of the greatest mathematicians who ever lived and accomplished much more sophisticated feats than formalizing integrals.

It should be noted, however, that not all integrals are compatible with Riemann's work with sums. But that is a more advanced topic.
• What is that big "E", why does he call it a sum, what is all of the notation around it, and if Sal has done that in another video, which one is it? What's it called?
• That "big E" is actually not an "E." It is the Greek letter "Sigma," which is the equivalent of the English letter "S." It stands for "Sum." If you can begin to see it as a "Sigma," or especially as the letter "S," the notation for integration will begin to make a little more sense.
If you are going to study more advanced mathematics (or especially physics, which uses this math), it would be advantageous for you to become familiar with the Greek alphabet, both the upper and lover case letters.
Hope this helps.
• What will happen if we take the limit as delta x approaches 0 ? Won't it give us the exact area?
• Yes! - and this video is meant to give you the background intuition for the subsequent videos where that limit will be taken giving a more formal definition of the definite integral.
• At , he says that x sub i-1 is for the left side, how should it be written if you want to use the midpoint or the right side?
• For right endpoints, the notation is simply x sub i,
left endpoints are, as Sal points out, x sub i-1,
midpoints are x sub i* (star), which looks like x sub i with a star directly above i.
• Why not just make it the sum of f(x) when i=0 to n? Since the first term is x sub 0. obviously the way he did it works with i-1 but it just seems more complicated.
• if you started at i=0 then you would have to do away with the "i-1", If we did things that way, we would need to change the upper limit of the sum to n-1 to ensure that we didn't take the height of the rectangle n+1 (which would have a height of f(n)). We could choose to change the upper limit but doesn't really capture the idea of a complete domain form a to b as well as 1 to n.
• Why don't we use simple integration method to find the area?
(1 vote)
• This section on Riemann sums covers the theoretical background that led to the development of integration. In this section you should gain an appreciation of what integration is by understanding the concepts of how it came to be.