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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