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Find the area of the shaded region.(Graph cant copy)

$$1$$

Calculus 1 / AB

Calculus 2 / BC

Chapter 6

APPLICATIONS OF THE DEFINITE INTEGRAL IN GEOMETRY, SCIENCE, AND ENGINEERING

Section 1

Area Between Two Curves

Integrals

Integration

Applications of Integration

Area Between Curves

Volume

Arc Length and Surface Area

Missouri State University

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Lectures

01:11

In mathematics, integratio…

06:55

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05:23

Find the area of the shade…

07:46

01:15

Find the area of each shad…

01:16

01:04

01:51

02:40

02:35

03:09

07:35

you're being asked to find the area off this shaded part in green, and you're given a few pieces of information to help you out. So first of all, you know that the sights of this triangle inside this circle, the sites are too. And you know that this angle right here is 120. So one thing that you could don't is remember that the area the whore circle this pie are square. So if you were if you were being asked to find the area of the circle, let me just write this area of the circle. It would be this, but remember, you don't have the whole circle if you divide the holster cost 360 degrees. But you only want 100 20 part of that. So you only want 1/3 off your circle so I can assure you. So that's wondered. And this that would be 2/3. And then there will be treaters. So you only one want dirt off it. So then your area for this whole park and ah, from here all the way there, including the triangle, it would be 1/3 pi r Square. So let me write area off the dirt off the circle. Images make this clearer here so that you understand where I'm trying to go. So the area Oh, 1/3 of circle would be wander Hi, our square again because the 120 degrees that means that you're gonna fit that three times into the circle. But you only want 1/3 off it. So the radius, we know it's too. So this is going to be 1/3 hi for two squared. So this becomes four. Third's by now. You would be done if they were asking you for the area off that part of the circle. But they don't want you include this little piece, this little trying. So then, toe, find the area of the shaded region. You would do the area off here in blue. Just put this one. So the area, well, your little point of the 1/3 off the circle minus the area off the triangle. This triangle that we have here, How do I find the area of this triangle right here? Because I don't want I don't want to include that part in my area again. You have the formula for the area, which is 1/2 part aside a which is to times Side B, which is also to time sign off 120 degrees because that's the tango in between. So and remember the area off the whole water. Third off the circle IHS for over three pie. So you're gonna take away that piece of the triangle. So if you put this into the calculator, you get a 4/3 pi my nest. This is 1/2 times two times two. That's just it'll and sign of 120 this radical 3/2. Now notice that you can cancel out this tools so you're left with 4/3 pi minus radical three, and then you are going to put that into the calculator way. This war over three times pi minus radical three and you're going to get the area off this. Let me show it off. This part in green Toby close to to a point five square units again to repeat, repeat the process. We're looking for this area and green. The first thing that I could think office find the area of the whole circle, which would be pi r square. But I don't want the holster Co. I want only 1/3 off it. That's how I got the Wonder Pyre Square, which is for pie or for over three pie. Once I plugged it into the formula. The next port iss. I don't want the whole section. I don't want the third off the circle. I wonder of the circle minuses triangle because I don't want to count the triangle. So the area of this triangle, it's one huh? Times your side, which is two times your other side, which is also two times sign off the ankle in between, and the and going between us 120 you're gonna simplify, and then you're gonna put it into your calculator, and you should get 2.5 square units.

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