Calculus I - Definition of the Definite Integral
Riemann sums help us approximate definite integrals, but they also help us between the representation of area as a definite integral and as a Riemann sum. Most likely, you are confused by the difference between the definition of (definite) integration and the most common technique for (definite) integration. 1. by reviewing finite sums and the relation between sums and integrals. . the lower Riemann sum, and these rectangles together will surely lie inside your.
This method was later used in the 5th century by Chinese father-and-son mathematicians Zu Chongzhi and Zu Geng to find the volume of a sphere Shea ; Katzpp.
The next significant advances in integral calculus did not begin to appear until the 17th century. Further steps were made in the early 17th century by Barrow and Torricelliwho provided the first hints of a connection between integration and differentiation. Barrow provided the first proof of the fundamental theorem of calculus. Wallis generalized Cavalieri's method, computing integrals of x to a general power, including negative powers and fractional powers.
Newton and Leibniz[ edit ] The major advance in integration came in the 17th century with the independent discovery of the fundamental theorem of calculus by Newton and Leibniz. The theorem demonstrates a connection between integration and differentiation. This connection, combined with the comparative ease of differentiation, can be exploited to calculate integrals. In particular, the fundamental theorem of calculus allows one to solve a much broader class of problems.How do you convert a Riemann sum to a definite integral?
Equal in importance is the comprehensive mathematical framework that both Newton and Leibniz developed. Given the name infinitesimal calculus, it allowed for precise analysis of functions within continuous domains.
Definite integral as the limit of a Riemann sum
This framework eventually became modern calculuswhose notation for integrals is drawn directly from the work of Leibniz. We have n right over here. The larger n is, the better an approximation it's going to be. So his definition of an integral, which is the actual area under the curve, or his definition of a definite integral, which is the actual area under a curve between a and b is to take this Riemann sum, it doesn't have to be this one, take any Riemann sum, and take the limit as n approaches infinity.
So just to be clear, what's happening when n approaches infinity? Let me draw another diagram here. So let's say that's my y-axis. This is my x-axis. This is my function.
Definite integral as the limit of a Riemann sum (video) | Khan Academy
As n approaches infinity-- so this is a, this is b-- you're just going to have a ton of rectangles. You're just going to get a ton of rectangles over there.
And there are going to become better and better approximations for the actual area. And the actual area under the curve is denoted by the integral from a to b of f of x times dx. And you see where this is coming from or how these notations are close.
Or at least in my brain, how they're connected. Delta x was the width for each of these sections. This right here is delta x. So that is a delta x.
This is another delta x. A reasonable way to conceptualize what dx is, or what a differential is, is what delta x approaches, if it becomes infinitely small.
So you can conceptualize this, and it's not a very rigorous way of thinking about it, is an infinitely small-- but not infinitely small delta x, is one way that you can conceptualize this. So once again, as you have your function times a little small change in delta x. And you are summing, although you're summing an infinite number of these things, from a to b. So I'm going to leave you there just so that you see the connection.
You know the name for these things.
And once again, this one over here, this isn't the only Riemann sum. In fact, this is often called the left Riemann sum if you're using it with rectangles. You can do a right Riemann sum. You could use the midpoint.