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Point) Suppose R is the shaded region in the figure. As an iterated integral in polar coordinatesf(x,Y)dA =E"I" f(r cos(0), sin(0)) r dr d0with limits of ...

Question

Point) Suppose R is the shaded region in the figure. As an iterated integral in polar coordinatesf(x,Y)dA =E"I" f(r cos(0), sin(0)) r dr d0with limits of integrationA =C =(Click on graph to enlarge)

point) Suppose R is the shaded region in the figure. As an iterated integral in polar coordinates f(x,Y)dA = E"I" f(r cos(0), sin(0)) r dr d0 with limits of integration A = C = (Click on graph to enlarge)



Answers

Let R be the region described. Sketch the region R and fill in the missing limits of integration. $$ \iint_{R} f(r, \theta) d A=\int_{\square}^{\square} \int_{\square}^{\square} f(r, \theta) r d r d \theta $$ The region inside the circle $r=1$ and outside the cardioid $r=1+\cos \theta .$

Do you want to find? Um, the area of the region bound by the core diode R is equal to two. Linus are two times one minus sign of data. Okay, so if you wanna go ahead and graft this, we can use it. Well, any software are by hand, Uh, basically ends up being this sort of upside down. Ah, cardio oId with a cussed right here. So basically ends up looking like this. This heart shape, um, blow. So this goes from 0 to 2 pi. Um, so we can go ahead. And, um, and your eight this from 0 to 2 pi. Um, So you want to find the area? Uh, is just The Dublin took over, Harvey. A go ahead and great district. Your two pi of and ah, from 0 to 2 times one minus. Sign data of ah. Already already data giving us Ah, integral from 0 to 2. Pi of, um, we have our squared over two. Is this integral? US. We end up with two times. One sign. Um, wanna send data squared all over two and a you change colors here. 1/2. Errol Flynn. 0 to 2 pi of this is, uh, four times that squared, Um, which is, um, one minus to sign data plus science. Great data. Do you data? And then go ahead and pull out. Um, this four to get two times in a row from 0 to 2. Pi of one minus two. Signed data Put a science. Where? Data data. Now, um, we can go ahead and start solving the 1st 2 parts very easily. But this part here science course data. Um, we have to use a You, um we can use reduction formula. So the production foreman, that would be, ah one minus co sign to date over to where this is equal to signs. Great data. And, um, we can go ahead, integrate that to get, um, our answer so we can go ahead and change this to the following one. Minus two scientific data. Um, plus 1/2 minus. Co. Sign to data divided by two. And this is all included in the data. Now we can ah, start integrating living us with, um two times data minus R plus to co sign data plus 1/2 data minus sign tooth data over to from 0 to 2. Pi and since co sign a sign of periodic These, um can't sell out when you go from 0 to 2 pi and we're just left with two times three have stada from zero to pi or two times 3.5 times two pi or, um, can't sing out these twos lives us with six pie as our final answer.

Do you want to find the ah area between the car e r E r Z equals one minus coastline data and the circle are they're gonna want. Now we can go ahead and draw us in the X Y plane. We know the circle Radius one is pretty simple. Her dogs is this. And if you look at our all our a minus one co sign data, we'll notice that it has a max radius of two, and a minority is zero. And since this one minus goes on that it's gonna be directed in this direction with our cusp here and our core diode poking like that. So we have to find the area inside the cardio aid and the circle. So we want to find, um, this area me go ahead and draughts feel better. Didn't blue. We want this inside area, so that is going to be a little bit, um, interesting to you all. We're gonna have to end up, um, splitting are and grew up if it'll help. And because we noticed that we have the intersection points at the top of the bottom where, um, what switches from negative or its which is from, um, cardio, it being the outside in the Carter went inside. So we're gonna have to, um, split it down the middle. China blue one after split in the middle. And they're not entirely accurate, but we have toe go ahead and calculate. Um, here, Here. We want to calculate this area in this area, so we'll notice that this left area just have a circle. It's just a semi circle. Um, these points aren't exactly ah, a line, but it should. It should be a semi circle of radius one. So this left area, um, we can use a double integral. Um, it would just end up being the double integral of, um, will have go from Native Pi. Over two are it Should be. Will go counterclockwise. We go. Pirate too. Three pirates too. Ah, from 0 to 1 of already rt data leaving us with 1/2 um, in a row from power for 2 to 3. Pi over two data. And as we expect, it should leave us with, um we end up with this is equal to pie. So we end up with pi over two as our area of our semi circle with Regis one and then we want to find the right area, which is a little bit more tricky. So the right area is the right area. Is these two humps of the cardio aid which we can find So we so from native parts you with a pirate too, of this cardio. So we'll go from we'll do the negative pyre or two prior to and its euro too. One minus co signed data of already already data leaving us with the interval from negative fire Two pi over two of one minus co signed data squared and 1/2 comes outfront all of the data. And if you go ahead and solve this, you can expand this real quick to get one minus to cover science data bless co sign square data and using our convergence cosign scored data is equal to one plus co sign to data all divided by two. We can go ahead and rewrite this as 1/2. They were all from negative tired to pyre to of ah, three halves minus two coastline data um plus co sign tooth data divided by two all of the data. And now, um, we can go ahead and integrate this easily to get 1/2 Ah, three halves data minus, um to sign data plus sine two theta over four. From negative pyre to to pirate too. We'll know that this is not going away studying this one we get no sign of. So we get at the upper bound, we get minus to sign A. However, to which is minus two and the negative pi over two, we get, um, minus sign of negative pirate too. Or bring NATO from we get, um, positive so we could go ahead and start evaluating this. So we end up with three halves times pi. It's time state up, which is three halves times pi. Then we get minus s. We get minus sign piratey, which is minus two and then minus two. Yet Appel up plus signed pirate who are split signing a pirate too, which ends up being minus sign power to are also minus two. And then we'll get, um, sign of pie inside of negative, I So that steals away, leaving us with 1/2 times 3/2 pi minus four. Or we end up with, um, 3/4 pi minus Who so it's our right area and then everyone add up are left area We end up with us of our total area is just pi over two plus 3/4. Pi minus two war. Uh, adding this this gives us five pi over four minus two as our area.

Which is equal Teoh. So the sign of pi over three is equal to the square root of 3/2 and the site, the co sign of pi over three is equal to 1/2. So this is equal to two pi over three over. Two times 2/1 are twos will cancel out on this is equal to the square root of three. So our final answer is going to be four Popeye over three, minus the square root of three.

Okay, so going ahead. And, um trying to find this, uh, iterated integral over our Where are is this Ah, inside the Lima song, Uh, R is equal to one plus 1/2. Ah. Co sign data. Now we can use our, um, graphing calculator toe graft this or or we can do this. Oh, analytically. We can go ahead and start going through different. Our data's are different data to find the are and we'll find out Data zero. You get ours equal to their house. That data's pirate. This pirate too. You get already won data. 33 days pie, You get ours, you gotta 1/2. And that they does. He could've three pi over to you end up with ours equal to one. And then once you get the two pi, it's like a little over again toe rz what they have. So we go ahead and have this sort of lopsided be muscle eso we have at three halves here. So have here. Then it goes to one. Then it sort of flattens out over here and goes back. So it's not a circle. It's more like a circle with a lopsided. Ah, Lopsided side where it goes vertical. And in this case, um, it's pretty easy to write it up. One of girls we have, we know that this goes from 0 to 2 pi we have, um let's say we double integral over our g of our data on the A. Go ahead and write this in polar cornets we had This goes this whole. My song goes from Syria to pie and data and it goes from zero to its maximum of one plus 1/2 co signed data and then our function is g of our data. Energy is already RD data. So if we go ahead and convert all this, this is a final answer that we yet for this problem


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