How do you convert a Riemann sum to a definite integral?

0:25 // Definition of the Riemann sum
0:50 // What you need in order to use a Riemann sum to find area
1:09 // The difference between a Riemann sum and an integral
1:34 // Converting a Riemann sum into a definite integral
2:09 // Converting an example Riemann sum into a definite integral
2:35 // Summarizing how to use the Riemann sum versus the definite integral

We know that Riemann sums estimate area, and we know that integrals find exact area. But how do we convert a Riemann sum into a definite integral?

The simple answer is that we just use an infinite number of rectangles to find the area, instead of a finite number of rectangles, like we normally would when we're using a Riemann sum. If we want to express the fact that we're going to use an infinite number of rectangles, we need to add a limit out in front of the integral as n (the number of rectangles) approaches infinity, and then the Riemann sum has become a definite integral.

As long as we know the interval over which we're trying to find area, we can change the limit and summation notation into integral notation, with the limits of integration reflecting the interval we're interested in. We change the function f(x_i) into a function f(x), and we change the delta x into dx. And that's actually all it takes to change the Riemann sum into a definite integral.

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Hi, I’m Krista! I make math courses to keep you from banging your head against the wall. ;)

Math class was always so frustrating for me. I’d go to a class, spend hours on homework, and three days later have an “Ah-ha!” moment about how the problems worked that could have slashed my homework time in half. I’d think, “WHY didn’t my teacher just tell me this in the first place?!”

[danzevr]

Now how would you change it to a definite integral format when you are NOT given the restrictions?

[hamsack]

I like the new format of your new videos.

[beereaters]

That was a great video to highlight the similarity (and, well, equivalency) of those two ideas. I know for a lot of kids that notation and jargon become incredibly confusing and act as a barrier to making the leap from approximation to calculation.

[amy_muk]

I'm surprised at how easy the conversion is!! Thank you so much for your simple but thorough explanation!

[zombiesandguns]

Miss, you're videos are the right approach. Thanks a ton

[ShooshProductions]

Excellent video! As a student who took Calculus 1 in high school coming back to take Calculus 2 nearly a year later in college, this video helped me understand the relationship between the Riemann Sum formula and integrals. The periodic breaks from the written content with facial close ups allowed myself to not get too sucked into what was on the page and stay focused on what was being said. Thank you for this simple and effective instruction.

[ItsEwani]

I can't express how much pain this video just saved me. I have been stuck on a Quantum Field Theory question for hours upon hours at this point and went into a frenzy of constant googling until I stumbled upon this video. Never heard of a Riemann integral before but this 3 minute explanation just helped me solve the problem in 10 minutes. The only thing I regret now is spending so long on the question in the first place...

[nealaudreysandiego]

Your style of teaching is very helpful, it's way easier to understand thankssss

[daswubs]

this is one best math videos I have ever watched. Very high production values. Pleas keep it up. It was very helpful

[dinohudson]

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[aliasjad]

Wow superb what a great method to understand ...good job mam

[timothymcbearington]

love it videos keep them coming ..helping me get through cal

[jasperjohnson]

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[oskarzdrojewski]

how do I proceed if I have dx^2 or dx^3 at the end?

[ptyptypty]

you are so ORIGINAL!! by that I'm referring to your STYLE of Presentation!!.. lol... this video was COOL as all GET OUT!.. :) thank you.... Krista King the Queen of Originality!!... You ROCK!

[ankushjha]

Thanks Mam it helped a lot to clear my doubts

[neocruz]

okay sohow did you get five, like how exactly do you turn it into one

[nroblly]

thanks for the video it was super helpful :)

[Redheadsmiles]

My teacher put a line over both x values in the parentheses of the Riemann sum and I don't know what to make of it or what to do with it, any ideas?

[pack]

You're a queen! I mean a king...wait...you're a goddess!