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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 $.

-64 and 16

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Missouri State University

Harvey Mudd College

University of Michigan - Ann Arbor

Idaho State University

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.

University of Utah