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# Worked example: Rewriting limit of Riemann sum as definite integral

When given a limit of Riemann sum with infinite rectangles, we can analyze the expression to find the corresponding definite integral.

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• If we are working with Riemann sum from the left side, should we write the definite integral from i=0 to n-1 ?
• Yes, this is correct. We are taking the value of the function on the left corner of the rectangle, so on the first time, we do not add a delta-x.
• At , Sal says that it's a right Riemann sum. How do you know based on the formula that it is a right Riemann sum?
• Because the sum has i = 1 and not i = 0 you are starting on the right side of the first rectangle as the first side would be located at i = 0 and the sum Sal uses starts at i = 1.
• Could we not also define the integral as the limit as delta-x goes to 0 of the general Riemann sum (in addition to doing so by setting as the limit 'n goes to infinity')?
• How does he know it isn't an integral with a bound of [0, 5], and the function itself is ln(2+x)? In this case, delta x still equals 5/n, but xi = 5i/n. An integral calculator gives the same result for both of these. Does it not matter which one you pick?
• So... how do we solve definite integrals for actual values other than using anti-derivatives? (i.e. can we plug infinity in for n and solve for an actual value other than 0? )

Because n is in the denominator of a term, and n approaches infinity, that value would approach 0. But maybe I'm just confused, very possible.
• Yes, that's the idea.
As 𝑛 approaches infinity, the number of terms also approaches infinity, while the value of each term approaches zero.
• What if delta x was (5/4n) would that affect the upper bound?
Also what if the function does not have an 'a'. what then? would the lower bound be zero or one?
• ofcourse it'll affect the upper bound because "delta x = (b-a)/n ".
if delta x =(5/4n) then (5/4n) = (b-2)/n => b=(13/4)
Also if the function does not have an 'a'. Then the lower bound will be 'zero'.
f(a+ (delta)X_i) this will be written like this " f((delta)X_i) " where 'a' equal 'zero'

Hope i didn't Confuse you and Sorry for my bad English
• What is the area under that curve . How can i use all this to find the area ?
• The definite integral or the Riemann sum are both the area under the curve under specified intervals.
• What about if we didn't have something like 5/n as the width, where b-a is clearly 5.. What if we had 1/n as the width in this same problem, what would we do then?
• Width tell us about "how far the upper boundary to lower boundary" over as many as N rectangles we want to use from our integral; (b-a)/n. if we have 1/n as the width then b-a or the "distance" from our upper boundary to lower boundary should be equal to 1 or b-a=1
• Recreating this riemann sum in software like symbolab or volframalpha, it interprets the limit of this integral as [a,b] = [0,5] instead of [2,7]. Why is that?
The same problem occurs in my textbook where a very similar sum to the one in this video appears, and you are supposed to express it as a definite integral:

*lim(n->∞): Sigma[i=1, n] (2/n)(ln(1+(2i/n)))*

Here, following the steps in this video we should get the interval [a,b] = [1,3] right? However the correct answer according to the textbook is [a,b] = [0,2], using software also gives me this same result.

I have been struggling with this for very long and I feel very confused about what the correct method of doing this is. What should I do about this?