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Prove (8) by (a) an area argument, (b) using Riemann sums and (c) using the Fundamental Theorem.

Calculus 1 / AB

Chapter 5

Integration and its Applications

Section 6

The Definite Integral

Integrals

Harvey Mudd College

University of Michigan - Ann Arbor

University of Nottingham

Boston College

Lectures

05:53

In mathematics, an indefinite integral is an integral whose integrand is not known in terms of elementary functions. An indefinite integral is usually encountered when integrating functions that are not elementary functions themselves.

40:35

In mathematics, integration is one of the two main operations of calculus, with its inverse operation, differentiation, being the other. Given a function of a real variable (often called "the integrand"), an antiderivative is a function whose derivative is the given function. The area under a real-valued function of a real variable is the integral of the function, provided it is defined on a closed interval around a given point. It is a basic result of calculus that an antiderivative always exists, and is equal to the original function evaluated at the upper limit of integration.

03:42

Riemann sums for constant …

03:46

Use the definition of the …

08:19

Riemann sums for linear fu…

03:33

02:21

(a) Write a Riemann sum ap…

let's start by drawing a quick, quick graph or for what? Our graph it looked like. So we have f of X equals C. So what? That means that if we draw our graph, our function will look like this where this is a constant line at sea and let's take a portion from A to B. So the one thing about this is that regardless off whatever XK star you use, you always have f equals C. So which means that regardless of you you were using the right some the left, some or the midpoint some this will always happen. Okay, so let's just keep that in mind. Now we will write what I was Some is so some is the summation and say we want and petitions. Okay. Our some is K equals one to end off f que star times Daughter X. Okay, that is pretty standard. Now, how do you get Datta ex while doubt to find out her ex, you take the length off your domain, which is B minus a divided by the number of petitions You have so end. Okay, now what is this whole Some equal. Okay, now what? You essentially doing here is that you are adding up all of the all of the f que stars together and you adding up and of those So each of these columns has value, see? And because there are and of these columns, then the then the total sum of them is see Time's n So therefore, the some equals C times n times Our daughter X here, which we found was B minus a. Okay, the ends cancel out and we get C times B minus a. Now what is B minus? What is C Times B minus a. What? Turns out that if you consider this area here this bound, it turns out that it describes the area off this rectangle because here this length is B minus A and this height iss see in the area it's just seat times B minus eight. And that's exactly what we have. So for a constant So so so for constant function, regardless off what? Off off What f Starr kay we use we always We always have the same. Our answer will always be the same. Exactly the same. It's the area under the curve

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