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Approximate the indicated area by connecting succesive $y^{-}$ values on the curve by a straight line, resulting in trapezoids.Show that the approximation of the area obtained by using trapezoids is equal to the average of the approximations using the left and right endpoints.

Calculus 1 / AB

Chapter 5

Integration and its Applications

Section 4

Approximation of Areas

Integrals

Missouri State University

Campbell University

University of Michigan - Ann Arbor

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.

02:03

The area under a curve is …

04:12

Find the approximate area …

Okay. So, uh, how to approach this problem? Let's start by drawing a curve here. Ah, so say the curve will look something like that. Ah, and maybe I'll dry twice. I'll drive. I'll draw the same cove again. Ah, so try to make it the same on Duh. So one of these, we were estimate using an inscribed rectangle. Remember? We're trying to estimate the area inside the coat on one of these will estimate using a circumscribe rectangle. Okay. Ah. As you can see, when we estimate using an inscribed rectangle we have Ah, we haven't underestimate on the stuff I shaded in black basically will be. The year will be an area. That Majesty error on here would have an underestimate. Okay, Um, so this is an open. We have an overestimate with this. Ah, with an error that's shaded in black over here, on the right on that. You can see one of these is in overestimate. At one of these is an underestimate. Ah, And if we take the mean of these two rectangular areas, we will we will cancel out some of the overestimate with the underestimate. And hence, this is why we shall expect to get a more accurate estimation of the true area and they need the code. Thanks for watching

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