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A table of values of an increasing function $ f $ is shown. Use the table to find lower and upper estimates for $ \displaystyle \int^{30}_{10} f(x)\, dx $.

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00:39

Frank Lin

Calculus 1 / AB

Chapter 5

Integrals

Section 2

The Definite Integral

Integration

Oregon State University

Harvey Mudd College

Baylor University

Idaho State University

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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A table of values of an in…

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Use the table of values to…

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Estimating a Definite Inte…

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The table gives the values…

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given. If we go ahead and plot these points here, we get the following and we're interested in finding the integral from 10 2 30 of f of X. We're estimating using upper and lower estimates. So we're going to use rectangles and it's up to us whether we want to use the left endpoint or the right endpoint. Let's just draw what would happen if we use the left endpoint here for each of these. So we would get the following rectangles on that at this point, would end up giving us a lower estimate because we have more negative area than positive area. So let's go ahead and write that the width of each of these rectangles, by the way, is four. Because it jumps from 10 to 14, 14 to 18 for the X values there. So I just wanna for all these on land now, we could just do length times with an animal up, or we could just factor the four out front and then go ahead and multiplied by all the left on points. So negative 12 and then minus six minus two. So we just combine all the heights, um, plus one and plus three. We don't use eight and the lower estimate, so that would be it. And then if we evaluate that, that will end up giving us a total area of 60. Sorry, negative 64. Okay, and then the upper estimate. Let's go ahead and do that just to draw that out. Let's go ahead and use. Oh, I don't know, maybe Blue this time, so that would start on the right side and go up and over. So we're going to use eight as a height three of the height, one of the height than negative to as a height. Let me see if I did that right? Yep. And then negative six as a height. So those would be the blue Rectangles would be the upper estimate there because there's more positive area here. So factor of four, because each of the rectangles has a width of four and then starting on the right side. This time we're going to add everything except negative 12, because that did not determine the height. So eight plus three, well, swarm and then minus two and minus six and simplifying all of this. This will end up getting us positive 16. So in the end, we know that the true area lies somewhere in between negative 64 positive 16 since those of the lower and upper estimate.

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