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(a) Let $A_{n}$ be the area of a polygon with n equal sides inscribed in a circle with radius $r .$ By dividing the polygon into n congruent triangles with central angle $2 \pi / n,$ show that$$\mathrm{A}_{\mathrm{n}}=\frac{1}{2} \mathrm{nr}^{2} \sin \left(\frac{2 \pi}{\mathrm{n}}\right)$$(b) Show that lim $_{\mathrm{n} \rightarrow \infty} \mathrm{A}_{\mathrm{n}}=\pi \mathrm{r}^{2} .[$ Hint: Use Equation 3.3 $.2 .]$

a. $A_{n}=\frac{1}{2} n \cdot r^{2} \sin \left(\frac{2 \pi}{n}\right)$b. $\lim _{n \rightarrow \infty} A_{n}=\pi r^{2}$

Calculus 1 / AB

Chapter 5

Integrals

Section 1

Areas and Distances

Missouri State University

Campbell University

Harvey Mudd College

Idaho State University

Lectures

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Okay. Hi. And this problem, we have a circulates Inside the circle is appointment of equal size, which can be by this. This is a center of the circle. This parliament is divided into a number of equal triangles. For example, this is this is one of this is one thing This is the next is the second language. This is the same triangle and support string this is this site is or And this site is also which is a sent a radius of the second. This angle is the central anger which is off. If you want to get the area off this triangle, we'll be acquitted. Video, I hope deploy. But Roy Oh, off deploy swing is equal to boy over. And where n is the total number of triangles in sizable in this case with before so But we want to get the area of the overall volume. So you way will not deploy this area by end, which is a number off trains it so in which, which is a total area of the Taliban with the equal or squared Uh, sign Do boy deploy, which is the first requirement, is the second requirement. So we we knew that the area of the bargain will be equal and it will deploy or square deployed and oy, sign to over in. We want to take limit to this equation so limit from then turns to infinity. Half are square is a constant eyes is very circus constant So it will be equal hope squared limit when tends to infinity Oh, look it sighing to over in We will take this equation and multiply it mainly by one. But in our case it will be too boy over do for So this limit will become I will show in exploit to will deploy and over to boy must deploy sewing Cool boy boy over in if we talk If we take this turn on and make it a diminished denominator every year it would become equals to guard no deploy toe over and over to boy over in Let system be seated which is are some thrilling So in the next flight limit when end girls 24 in will be equal Oh squared deploy limit when sita zero Because what we're increasing the number and the sita or the central angle will decrease gradually until it nearly become zero Flemington. See? It tends to zero toe boy Want deploy saying see you seek mint. We will take the two boy outside so exploit limit and well in turns to infinity will be acquitted. Pool off, squared, deploying to boy Look, Chloe, limit tens zero So I'm so in Saint Overseas. The tour Big gun with the two. So it will be equal. Sister is one. This time would be equipped to one because we see the term is 20 signs. It was one through This time is one. So this term will be equal next light limit and then turns to infinity. Boy R squared one. She could Boy. Oh, squish. This is a quote on This is the overall area of the city. This is the way as they used toe proof as overall area of the second as shown, uh, isn't it you? Thank you.

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