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A) Set up (but do not solve) an integral to find the areainside r=3 cos θ and outside r = 1+ cos θb) Find the polar point when the y-coordinate of theparticle...

Question

A) Set up (but do not solve) an integral to find the areainside r=3 cos θ and outside r = 1+ cos θb) Find the polar point when the y-coordinate of theparticle is -1 in the interval 0≤θ≤π/2c) 7. Find the area inside r2 = 4sin2θ

a) Set up (but do not solve) an integral to find the area inside r=3 cos θ and outside r = 1+ cos θ b) Find the polar point when the y-coordinate of the particle is -1 in the interval 0≤θ≤π/2 c) 7. Find the area inside r2 = 4sin2θ



Answers

Use a double integral to find the area of the region.

The region inside the cardioid $ r = 1 + \cos \theta $ and outside the circle $ r = 3 \cos \theta $

All right. We want to find the area inside the cardio. I'd articles one plus coastline data, but outside the circle R equals three coastline data. So here's what articles one plus coastline data looks like in rectangular. So we'll use that to get the picture in. Polar. It's at zero degrees were at two at five or two were at one. And by the time we get to pie, we're back. We're at zero and all in a positive way. And then the same thing on the other part. Okay, that's pretty good. What colored in? All right. And then articles three Coastline thing that looks like this at zero degrees, we're at 3123 and then at pi over to work to zero and then back again, Pretend like that's a circle. So we went inside the cardio it but outside the circle. So this green part right here. Okay, so it's easy to see that we're going from the cardio oId to to the, um, circles. So from the card oId to the circle, our it goes three coasting data R d r d theta. So the interesting problem here is what angles are these from where to? Where should we go? So we need to set the two equations equal to each other, since they're both equal that are so one plus cosine theta equals three cosine theta. So one equals to cosign. Data go. Xanterra equals one half. So coastline 30 equals one half, one half square 23 That's 60 degrees, or pi over three. So let's call this one minus 60. And this one positive 60 or minus pi over three two pi over three. So that's the integral. All right, so first integration gives us our squared over two. So let's put one half once by over 3 to 5 or three r squared from one plus coastline data to three coastline data geetha. Right now. Okay, so it's one half in a grove minus pi. Over 3 to 5 or three. Nine. Cosine squared. Data minus one plus to cosign data plus coastline square data the photo. So one half minus five or 3 to 5 or three. Eight. Coastline squared. Data minus two. Cosine theta minus one. Did they know? Okay, so we're gonna put in an identity for that coastline squared. So I got one half when It's Piper 32 pi over three. Eight times one plus cosine. Tooth data over to when is to cosign theta minus one. Uh, data. I'm starting to really go sideways here. Okay, so see if I can straighten out to one half minus pi over 3 to 5 or three. So we have eight times a half, which is four for minus one three minus plus eight over to which is four. So four cosign tooth data minus two. Cosine theta d theta. Now I'm ready to integrate. Still going sideways. One half. What website back in here. One half, three fada plus to sign to theta minus two sine theta from minus pi. Over 32 pi over three. All right. Okay. We're gonna We're gonna need all four of them, so let's just put him in here. Square to three. It's where 23 minus the square to three. Minus the square to three. So I have one half. Three times five or three. So pi. Plus two times the sign of two pi over three, which is square to three over to minus two. Signed pi over three, which is squared of three over to minus parentheses. E three times minus pi over three, plus two times the sign of minus two. Power three screw three over. Two minus two times. This one. All right, so now we have one half pi plus pi. So two pi two squares of 3/2, minus two squares. Three over to those canceled minus. Plus those canceled E get pie for the answer.

Today, we're gonna be looking at how to find the area under the cardio oId using double integral Zell's and polar coordinates. So this is what this cardio it looks like, Um, his picture that I drew Andi, This is the carnivore that we have. Um and so as we can see that we have our a point here where it's starting at 00 and then our wise again zero at this point, and then it is continuing. And so this is helping us see all the points going around, and then it's going to the same point again. So we know that we want are are to be equal to zero in this case, Um, if you didn't have the ability to look at what the graph is, you could just be testing out some of the numbers. And seeing with the pattern is in order to find out what the boundary is, eso we're just going to solve this would have a negative one is equal to Negative Coasts state. Uh, and then we have two by two by negative one course, so we'd have one is equal to coasts data and coast is equal toe one at two points, Cose is equal to one at zero. And, uh, two pi. Um, so we know our data is going to be, um, greater than or equal to zero, and it's going to be less than or equal to two pi. So this is important. Remember, um, on DSO from here, we can start writing an equation out. So we have, um, our second part, which is going to be our theaters. So we have 0 to 2 pi on. We have to have our first part, which is just going to be from 0 to 1 minus coast data, which is the equation we were given equal toe are. And now we're gonna have our d r de data, and this is going to be our to our double integral that we're gonna use it order to solve for the area. So first thing I'm gonna do, just rewrite this brackets around apart that deals with our our variable, and then I'm going to just solve for our our integral. So again, just rewriting this. And now the integral of our is our squared over two. And we know that this is from 0 to 1 minus Coast Data de Seita. So now we can plug in our numbers and we have one minus coasts. Data squared over two minus zero, squared over to and all of this times de data. Um, and of course, this is just going to be equal to zero. So we're gonna be able to get rid of that heart. So we're just going toe Have 0 to 2 pi of one minus coasts. Data squared over two D fate. And we have to solve for this for someone to take out my one enough. Okay. And just re right this. And now from here I'm going to do multiply one minus coasts, feta by one minus coast sita de data. So we have one minus two coasts. Data plus coasts squared. ST. Ah, all this time, STI data from here, we can start splitting, um, the equation into separate intervals. So we'll have integral from Syria to pie of one d theta minus two times the integral 0 to 2 parts of Khost data D data plus integral from 0 to 2 pi of coasts squared theater di fada. Okay, now we can write down or went up again and we can actually just solve for our 1st 1 So first part, the integral off one di fada is just gonna be equal data from here minus two and the integral of co sign data data is just equal to sign. So we have two times Stein Seita from 0 to 2 pi. And now we have this equation which we can't yet just solved for. We have to rewrite this on. We can rewrite this as, um the integral from zero to pi over two of co sign to Fada, plus one over to Di fada. Reason why is there's a trick function that states that, uh, the coasts of two fada is equal to que coast squared data minus one. So if we just make some adjustments we see we have this equation right here. Okay, Andi, Now, from here, I'm just going to rewrite what we have. Waken also split our last integral into two into girls. So we're gonna have plus the integral from Well, we could take out a 1/2 1st Actually make our lives easier, said the integral from zero to pi it a two pi of co sign Teoh Fada plus the integral from zero to pi of one. So our Tuesday the deed data plus the integral from 0 to 2 pi of one de Theda And all of this is times that 1/2 and then times the other 1/2 Um And now we just have to do one more adjustment and we'll be ableto actually plug in values and solve all of our intervals. So we have the co sign of two beta. We're gonna do a U substitution. We're gonna say data is equal to you. So to De Veda is equal to d you. Di Fada is equal to d you over to Onda. We can plug that in now and we can solve for our last integral We're gonna say 1/2 of beta from 0 to 2 pi minus two times Sign data of 0 to 2 pi plus 1/2 times the integral from zero to two pi of coat sign you d you all that over to on? We're going to say plus so the integral of want it d fada, It's just gonna be data from zero to pie on. We have that market. Okay, so now we have half times beta from zero minus two times. Sign data from 0 to 2. Pi plus 1/2 and now weaken small for this, um so co sign of you over to the integral of that, it's just going to be thesis sign of you over to and that's gonna be from zero to do. Hi, Onda. We have data from 0 to 2 high All of that heart times in half, and then we have the greater 1/2. And now we're gonna have to substitute back in for are you and you is equal to two fada in case you forgot. Um, so I have 1/2 data from 0 to 2. Pi minus two times. Sign fada from 0 to 2 pi plus 1/2 times theist. Sign of two data over to 0 to 2. I plus data from 0 to 2 pi. And now we can finally start plugging in all of our values. So we have 1/2 times to pai, minus zero minus two times sign of two pi. Um, plus two times the signing zero plus one worth of sign of people at times chewed by over to minus 1/4 of sign of two times zero. There's not overtake. Sorry. We did. I did. The 1/4 are So we're good. Okay? Yes, we have. Ah, 1/2 times to the pie minus. Oh, what? Half times? Zero. Uh huh. Now, from here, um, we can start simplifying so we can cross that Are zeros, of course. And pie of to sign if I over two is equal to zero and sign of zero is also equal to zero. So these are gonna get crossed out. Sign of four Pipe is equal to zero on again the sign of zero Cause this is just going to be a sign of a few times. You're just going to be equal to the sun. Zero that's equal to zero sets gets crossed out. So we're just left with 1/2 times que pi plus pie. So this is equal to three pie over to and this is our area

Find the area of region close by thieves to cardio IDs. So I've drawn these two cardio IDs and red blue. And the idea is, instead of trying to do this in a more complicated way, we're just gonna take one of these. Find the area than multiply times for each one of these. Here is the same as each one of these here. It's not drawn to scale, but they're the same because of cemetery. All right, So to find the area of one of these sky and black here, we're gonna find it like this, right? We're going from zero to two pi and the smallest radius zero. And how our radius is changing is depending on this cardio it here. Okay, so doing this integration your own eye over too. One minus co sign Data squared. Yeah. So what we do here is we We multiply this out, we foil it out, and then we just integrate again, Right? So to integrate, this will have to write this as one half one plus co sign to data. Okay. And then you're going to do this integration, right? This is the sum of the work that you'LL have to do. But in the end, we'LL get one fourth times three five or two minus four. Now, this is just the area of one, um, peace. Let's call it peace. It's just the area of one of these, but we see again, we have one, two, three, four of them. Okay, so the total area is four times one over four times three I over too, minus four. So the area enclosed by these two cardio IDs is given by this.

Okay. We want to find the area enclosed by articles one plus coastline data and articles one minus coastline data. So first we need to draw them and to draw them, I need to draw them in rectangular first. All right, So here's pie. Here's two pi. Here's the regular cosign will go through the's five points. So if we add one to it, it will screw it up. One. We'll go here, here, here, here, Here. So it'll look like this. Kind of. So in polar at the angle. Zero, it's at two at the angle pi over two. It's at one. And at the angle pie, it's at zero. So there's half of the heart and then the other half, all right. Now let's look at the other one By 25 It was just the cosine. It would be this, but it's minus the cosine. So it'll be here. Here, here, here, here. And then we need to scoot those up one. So here, let's do a different color. Here, Here, here, here, here. Sort of look like this on time. Install the point. Okay to pretend. Okay, so had zero. It's at zero at five or two. It's at one. At pie. It's at two. So it's just a cardio. It going the other way. So there it ISS So we're looking for the area enclosed by them. So that would be this part right here. Okay. And even though you can't tell by my picture there symmetric So if we can just find the first one or even what's in the first quadrant that will fix it. Good. Let's see where they're intersecting. One plus coastline. Data equals one minus coastline feta. So subtract one from both sides and co sign to both sides. To cosine, theta equals zero. Hopes it not by not data hero. So data is, um, the pirate twos three pirate to etcetera. Okay. Okay, so we're gonna send our out. It's gonna go from here out to there. And so it's gonna hit the green one. Which was this? So zero thio, however, to 0 to 1 minus coastline data one, our DRD data, and the one is because we're just finding the area. Okay, that's gonna get us this section right here that I've colored in blue. So we're just multiplied by four. And I should give us the answer for zero pi over two. And a girl bar is R squared over two from 0 to 1, minus coastline data D theta. So bring this to out we get to zero pie or two one minus cosine. Theta squared D theta So 20 to Piper to one minus two. Cosine data plus cosine squared data. Do you data. Okay, so that's two data minus two. Sign data plus the integral. Yeah, we've got a coastline squared here, so you have to put in the identity one plus cosine tooth data over to do you data. I'll wait and plug the zero and pi over two and everything at once. So we have to data minus two. Signed data plus one half data plus one half signed to theta. Oops. And all that. Zero pyre. Vertical. So have to five or two minus the sign of private to two times the sign of private too. Plus one half five or two plus 1/4 the sign of two pi over two, which is zero, my ass. Everything else is zero. So we had to minus here. We have a pi over two and a pi over four. So three pi over four. That's where this to come from. I don't know. Um, start again Right here. So we have, uh, two times pi over two plus five or four. So three pi over four minus two. So three pi over two minus four. That should be the answer.


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