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# Riemann sums in summation notation

Generalizing the technique of approximating area under a curve with rectangles. Created by Sal Khan.

## Want to join the conversation?

• At , he says "X sub i-1." What does "sub" mean?
• "Sub" just means that it is in the subscript, the smaller text following the bottom half of the X.
• Just wondering. How is this different from a right Riemann sum or a midpoint Riemann sum?
• The right hand sum is where instead of making f(x) the value from the left side of the rectangle, it's the right side. Midpoint is where you take f(x) where x is in between the left and right endpoints of dx.
(1 vote)
• 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.
• 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.
• 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.
• 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.
• 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 don't we use simple integration method to find the area?
• 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.