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(a) Use six rectangles to find estimates of each type for the area under the given graph of $ f $ from $ x = 0 $ to $ x = 12 $. (i) $ L_{6} $ (sample points are left endpoints) (ii) $ R_{6} $ (sample points are right endpoints) (iii) $ M_{6} $ (sample points are midpoints)

(b) Is $ L_{6} $ an underestimate or overestimate of the true area?

(c) Is $ R_{6} $ an underestimate or overestimate of the true area?

(d) Which of the numbers $ L_{6} $, $ R_{6} $, or $ M_{6} $ gives the best estimate? Explain.

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Calculus 1 / AB

Chapter 5

Integrals

Section 1

Areas and Distances

Integration

Oregon State University

Baylor University

University of Michigan - Ann Arbor

University of Nottingham

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.

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so this problem as us to use to use the rectangle message for different simple points to estimate the integral F from zero to twelve or the area. And there's a curse, which means in the group, if you have the tax book, you can see that the function is decreasing concrete down and in each case they ask you to use six rectangles. Therefore, each issue off them will have based too. Ah, base of lens too. And hideaways rectangle depends on the employees on the use. So in the case of the if you want to use the laugh and point than Zen Point will be actually ho zero, two, four, six, eight ten But and each issue off one has ish off them has silenced to a starbase. Therefore, your integral estimate will be like this zero. Plus, they're forced to cool. The last one left one point p f off ten for the meadow M point again each Each of them has a rectangular space too. And the middle Cemal point will be one three, five up to eleven. And I'm sorry, this is a middle cemal point. So this should be past three. And if you want to use the the right symbol point, which is part two. Then the end point will be two for six up to twelve, and you can figure out this value by looking at the crowd from your textbook. So because it's a decreasing functions and laugh and point is higher lands middle point, its hunger in his heart and the Ryan Point. So it's eleven point that's who overestimates area the right hand point Matthews. Because we take the lower one, we're on dress made area and and because is decreasing in the middle, the middle point is closer to the average off F elt That interval compared to the left or right on point, therefore gives up assess Nate.

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