Problem

Functions with absolute value Use a calculator an…

02:20

Problem 76 Easy Difficulty

Functions with absolute value Use a calculator and the method of your choice to approximate the area of the following regions. Present your calculations in a table, showing approximations using $n=16,32,$ and 64 subintervals. Make a conjecture about the limits of the approximations.
The region bounded by the graph of $f(x)=\left|25-x^{2}\right|$ and the $x$ -axis on the interval [0,10].

Answer

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Calculus 1 / AB

Calculus 2 / BC

Calculus: Early Transcendentals

Chapter 5

Integration

Section 1

Approximating Areas under Curves

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Video Transcript

So if you type into Google does most Riemann sums, it will bring you to this program. You have to click on the program. But now I could just type in the function that they give you, Uh, this 25 minus x baird. And you probably noticed I hit this absolute value key to make that appear. And they also tell you that we Onley want the interval from zero to 10. So let me zoom out so you can actually see this graph. Oh, man, I really had to zoom out because of that. 25 Really, really accentuated everything. Um, and there's a lot of things you can do with this to you can change your settings. Um, so maybe you only care to see on your x axis from, uh I don't know. Let's make it like negative five to 15 on your X axis. I'm happy that it's going up to 60 because the graph goes up pretty high there. But anyway, the directions ask you toe make an equal 16 the number of rectangles. So now you can see that there's 16 rectangles in there and we want What do we want? Um oh, I don't see where it tells you if you want left, right or midpoint sums. So I'm just gonna leave them as left about that. Um, they're all gonna be pretty good approximations because we have a large number of rectangles. So by using this, uh, like if and I could click on all these values when x zero, we get a Y value of 25 that should make perfect sense. And then the next x value and we could see all of these. So if you're asked to make a table, you can use this program to find all of those. And if you eso each wit looks like it's 0.6 to 5 times each height, um, and add them all together you get about 234.375 So then the next interval that they want is to have 32 rectangles. Okay, so we get a even a better answer because you'd like to think that the areas above the curve will cancel out the areas that were missing below the curve. Um, what's the next one? 64 intervals. Now you can barely even go above you, barely even miss any below. So if you're looking, it looks like we're getting a value closer and closer to 246. And what's nice about a computer based program? We get a larger answer. 249 um, is a calculator won't give you like it takes forever to do all these calculations. But on the Internet, all of these answers pop up really, really quickly. So now that I'm adding more zeros in here, I'm convinced that this answer will be closer and closer to 250. So much so that I would be willing to commit my conjecture about the limits of these approximations. We're getting closer and closer to 250 so there you go.

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