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Evaluate the Riemann sum for $ f(x) = x - 1 $, $ -6 \le x \le 4 $, with five subintervals, taking the sample points to be right endpoints. Explain, with the aid of a diagram, what the Riemann sum represents.

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04:37

Frank Lin

03:31

Stephen Hobbs

Calculus 1 / AB

Chapter 5

Integrals

Section 2

The Definite Integral

Integration

Campbell University

Oregon State University

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.

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Okay, so we here we have the function F of x, which is x minus one. And I drew it on the graph, this is xy regular coordinate plane. And we're interested in doing a right endpoint riemann sum from minus 6 to 4 with five sub intervals. So first thing we want to do is find the width of our interval and in general it's B minus a over N. We're A and B. Are these endpoints? There's your A and there's your B. So that gives us four minus a minus six, All over five, So that's 10, 15 or interval width of two. Okay, so that means we're gonna break up by two. So um what I'm gonna do is I'm and then I'm going to make uh rectangles with right endpoints, let's start from there. Right? And uh the right endpoint is what's defining our functions, let's see if I can Draw straight enough lines. Okay, so the right endpoint height is at three because we're at the .43 and we're going to go over to because we're using the right endpoint, so that's our first rectangular for our approximation, then um This point is our next rectangle because that's our right endpoint and that's at the .21. So the heights one then the next height is determined by um uh this endpoint, So we get this rectangle because it's always by the right endpoint and then this rectangle. And finally this rectangle, we just need those heights. Okay, since f of x is x minus one, whatever the X value is, the y value is one less, so that's zero minus one, This is -2 -3 And this point is -4 minus five. Okay, so now we're going to do the Riemann sum for the right endpoint and that's just basically we're going to add up all the rectangles and if their below the X axis they do count as negative. So we will start from the right going left. So we have a height of three and a width of two. Then we have a height of one and then with the two Height of a negative one, and with the two they all have the same width of two, then we have a height of um The scene -3 and with the two And a height of negative five and a with the two. And when we add all that up we get that are Riemann cem Is equal to -10. And then the questions well what does that mean? We're only approximating. But basically if we're to find the integral from minus 6 to 4 of our function then we can say it's approximately not a very good approximation because those rectangles are really big But we can say it's approximately equal to -10. So that's the meaning of what we're doing. So hopefully that helped have a wonderful day

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