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Graphical Reasoning Consider an $n$ -sided regular polygon inscribed in a circle of radius $r .$ Join the vertices of the polygon to the center of the circle, forming $n$ congruent triangles (see figure).(a) Determine the central angle $\theta$ in terms of $n .$(b) Show that the area of each triangle is $\frac{1}{2} r^{2} \sin \theta$ .(c) Let $A_{n}$ be the sum of the areas of the $n$ triangles. Find $\lim _{n \rightarrow \infty} A_{n} .$

Please see explanation.

Calculus 1 / AB

Chapter 4

Integration

Section 2

Area

Integrals

Campbell University

Harvey Mudd College

Baylor University

University of Nottingham

Lectures

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to have the following problem. There's Ah, there's some pizza for you want to slice? I want to be fighting D'oh! And he calls license. Oh, so he's not perfect drying, but you already have. It wasn't that bad angle where the same for all of these pieces. So I go there way have some radius here. So that way we do the whole turn you get, Uh, Bye on then. Each of these lice is us. Mango feta on olders and of them and his laces. So you want to figure out you have those in his license and the radios? What is the area? My theory of that shape. You there? Well, uh, if you believe that his life was the same Oh, area. Uh, What's gonna be in times? It's a van. Where s event is the area of a single slice here? Not shaded area there. That's one of those pieces. A seven. So for his event, you have been following setting. So they're triangle. Do you have the sango? Huh? On the his distance, there is the same LaSalle. This knows there are all right. So so loose travel would Well, we can cut it in how unfolded to see that the dollar here, it's gonna be ableto Well, if you're full Did is gonna vehicle to the area. Piece tangle. Right. We're going for here. You have this angle. That's Hobbs. That is the same wood there hops. Well, if you do, we have these triangle one of those. Well, these your triangle is the same. Was that about over travel? So here we are. No. Perhaps you have are. But our here's be sense. So well, the area of these rectangle I was gonna be these distance. These those picks legal Does the snow's X here is that the snow is white. All right, extends. Why, that is the area office when eggs, That's why. But what is what is sex exists? Are was signed off the house. Our cause. Books? No, that is Baron. That is why are called on. Uh, that is way on eggs. Yeah. My butt on X is equal to our sign. Perhaps sign of her house. So, uh, also that if he did a product, you get the delivery is gonna be our squirt. We'll sign off their hooks, then sign off the rocks. Sign off in the house on the DL angle. Identities tell you that these gadgets, they're very simple Toe. What? Huh? Off sign off the that I'm going in their place. So one how both sign will be Sango two times two attacks. So that was very nice thing happens here. As you can guess, the two goes away, but and so you have area will still have the arts. Quarter square little area of these single piece will be of a single's life. It's gonna be our square loves sandal, but, uh Well, uh, what is it? Well, we have in his laces. So if there should be such that and that was that I go gives you the whole turn Did you buy? So we can be fired, but and wonderful. And they get this cancels so later we got to find a way to buy him. Great. So this area is abandoned on a vehicle toe our square replacing for what it is. Sign off to buy over in. So as we if it is drawing, we see that you Syria is cause trying to leave. Oh, it's not. It can be greater than here of the circle because he's constrained there so he can ask yourselves little area there. This deal is that how easily the same goes for you. Katie is early. Need us saying go to infinity. Hey. Hey, Here is that little area. So sometimes it's a band is gonna be ableto in terms. This so is their lead to that area, Which will? There should be vehicles. You got these constrained to stay within the circle. So what is the lead off? N times are square hubs square house. So, you know, bye over in. And if you just go ahead the Bailey quitting Cy Baby, will you get the beauty times are square house That was signed off. All the results to zero. This is zero, but this is zero. So serious times infinity. What is that? But I will when you lose, you can manipulate this and already does, uh, wanna burn so peace Can we read us? Her square huffs sino good bye over end. We can bring that What? Multiplying white and is the same musty by biting by Want over so that well you think you are I'm going Divinity. We're gonna have a local form, so it goes to zero zero or zero. Yeah, we're gonna blow people. So after a fine no pickle Hello? Get her in. Shape it up with respect to end. So you're The sign is that is what we have been through. They are square house people Sign is oh, sign, but well, the sign of the same But we have your depression with respect And so you get a factor of minus two by over a squared over since word And then we differentiate the bottom with respect when minus one Over in squared I said I'd Well, something wonderful happens here. I wonder where it ends where it comes to that so that well, you clean equals angle into beauty. You're gonna have to co sign off zero, because by wearing goes baby Mr Zero. So, cause I hope, Cyril, but he's gone. Okay, so So these whole expression is gonna go to grows through. Uh, so are square Hobbs. Them school sign of 01 times one times to buy them. Still buy. I do. But until another nice thing happens here to to to get that the leader is the area. Oh, the circle are scored in spite, but he's here, not circle. Cool

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