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If $$ f(x) = \cos x \hspace{10mm} 0 \le x \le 3\pi/4 $$ evaluate the Riemann sum with $ n = 6 $, taking the sample points to be left endpoints. (Give your answer correct to six decimal places.) What does the Riemann sum represent? Illustrate with a diagram.

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Riemann Sum $\approx 1.033186$

00:02

Frank Lin

04:46

Mutahar Mehkri

Calculus 1 / AB

Chapter 5

Integrals

Section 2

The Definite Integral

Integration

Missouri State University

Campbell University

Harvey Mudd 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.

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03:14

Okay, so we have a cool problem here, we have a graph of Y equals cosine of X. Or f of X equals cosine of X. We want to do left endpoint riemann sums and we're gonna do four sub intervals and we're going to go between zero and um oops, it's supposed to be six sub intervals, let's fix up. Um not four intervals but six intervals, let's fix that. Okay, so 66 intervals and then we're going to break up use those six intervals between zero and three pi over four. So first thing we want to do is find the width of our interval. So we're gonna do b minus A over n. Where this is A. And that's B. So that will give us three pi over four minus 0/6 intervals Or three pi over 24, which reduces to Pi over eight. So the width of every interval would be pi over eight. So let's kind of create this, we've got there we go. And so this will end here at three pi over four. Okay, we're doing left um endpoints, so let's see if I can draw this uh somewhat straight lines. Okay, so that means the height of our rectangle is determined by our left endpoint. So this would go up left endpoint determining it and the left endpoint determining the next one, trying to get him straight. Not perfectly straight, But the left endpoint is what determines each of our rectangles. This left Hand zero. So that's an easy one and then we have a left endpoint like this. So so we are going to 12345641 has zero area. Okay, so what we're going to do now is find our left hand Riemann sum and then we'll relate it to what it equals. Okay, so the left hand Riemann sum is going to be the height determined by left endpoint times the width. So we're going to do Kassian of zero Times the width of pi over eight plus cosine of pi over eight. Right. We're going in increments of pi over eight Times of width, apply over eight and so on would just keep adding pi over eight. So that's going to be two pi over eight, which is pi over four, Wits are always pi over eight and we still need three more. So then we'll do co sign of Um three pi over eight Times Pi over eight. And let's see the cosign of four pi over eight, which is pi over two, That's the 01 and finally we'll have cosign of 55 or eight as our last left endpoint and it all has the same way. All right. So when you plug that in the calculator, you end up getting 1.33186. So that's my left hand Riemann sum. Um And the meaning is or the question is well what's the meaning? The meaning is that If I were to take an integral from 0-3 pi over four of my function. Kassian of apps, then I'm approximating it by my Riemann sum, so it's approximately equal to one point oh 33186. Now, it's not a perfect approximation. If we had more and thinner rectangles, it would be better, but it's what we got with our six um six intervals. So hopefully that helped have an amazing day.

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