Problem 15 Easy Difficulty

Use a calculator or computer to make a table of values of right Riemann sums $ R_n $ for the integral
$ \int^{\pi}_0 \sin x \, dx $ with $ n $ = 5, 10, 50, and 100. What value do these numbers appear to be approaching?

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00:02

Frank L.

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Video Transcript

All right, We've got a question here. We've got the integral from zero. Hi, Synnex. Respect to Thanks. All right. And we want to use remind song to determine this value. We also want to use a computer and 75 10, 1500 And see what these values appear to be approaching a number of these values preparing people. All right, So we can start off by taking the endpoints of our sub interval. Start off with setting an equal to five, and we know it would be from zero and five. So you're changing X. Mhm. Excuse me. Zero umpires. You change it, Actually, pi minus 05 Fireworks. Fireflies. All right, so then your x of one is pi over five. And then your exit, too. Hi. See the two pi five. And it just goes on so on and so forth until you get to or accept. Yeah, if I was just £5 or five, which is just pie. All right. Here. What we're gonna do is we're gonna look too, get our mid points. So we see our mid points would be for first region from zero. You know, execute zero and pi or five, which would be pi over 10. And then we would have our midpoint from hire five and two pi over five. Which is the same thing has you can use my handy dandy captain contribute three piles attack and then you just go on and so on and so forth. So you calculate all your limit, you're midpoint values, okay? And then you can go ahead and solve for your Raymond some by taking the taking. The stylist is acting up right now. Okay. Seems to be working, so we would take our agreements down by taking it will change in X multiplied by the function of X sub one, the mid point of the you can accept one midpoint. So basically, you're looking at the values between zero and pie Oh five. And you go on and do the add to this up until you get to the point where you have the midpoint for your last Southern rule. This is just to clarify and write it all out. So it's a little bit more clearer is you're taking your change in X, which is pyro five, and you're just multiplying it by the mid mid point between this region I overturn. What's the midpoint between this region? Just three pi over 10. And you just keep doing it for each interval, so you'll have pilot two. Oh, I'm sorry. We got to write the function of X function pile to 10. What's the function of three Pile turn. What's the function of fly over to what's the function of seven pi over 10. What's the function of nine pi over time? All right. And then you would say, Well, what would be my function of pirate tent? So your function of X is actually this here, the sine of X. So you're going to say that you're gonna plug in that value for X and solve for that sign of X, You plug in pi over 10 4 sine of X, solve for it, and then software it with three pi over 10 pi over 27 pi over 10 and a 9.5 10. You would add those values altogether multiplied by your pie over five, and your value should come out to be 193 we 766 All right, now, looking at the remainder of this question, you see that it asks us to solve for it for when N is equal to 10 15, 100. That would be a lot of sub intervals for us to do by hand, so we would essentially plug it into a computer. And when you go ahead and plug it into either your calculator or your computer software, which everyone you're using to have that one N is equal to 10. You have 1.98 women. Summers, you could say 1.9835 to 4. When you plug in 50 you get 1.99 342 And finally, when we plug in 100 we'll get the value. 199836 2.999836 All right, And then, finally, the last part of the question is asking us, what value do these numbers appear to be approaching? And we can see that they're all approaching what looks like to be, too. It's a final answer is approximately. All right, Well, I hope that clarifies the question there. Thank you so much for watching