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A) x2 Y =1. Use Green's theorem to compute the area inside the ellipse 72 182 Use the fact that the area can be written as dxdy=i Jo ~ydx+xdy Hint: x(t) = 7 co...

Question

A) x2 Y =1. Use Green's theorem to compute the area inside the ellipse 72 182 Use the fact that the area can be written as dxdy=i Jo ~ydx+xdy Hint: x(t) = 7 cos(t)- The area is 126piFind a parametrization of the curve x2/3 + y2/3 12/8 and use it to compute the area of the interior: Hint: x(t) = 1 cos? (t)_ 6174n/5832

A) x2 Y =1. Use Green's theorem to compute the area inside the ellipse 72 182 Use the fact that the area can be written as dxdy=i Jo ~ydx+xdy Hint: x(t) = 7 cos(t)- The area is 126pi Find a parametrization of the curve x2/3 + y2/3 12/8 and use it to compute the area of the interior: Hint: x(t) = 1 cos? (t)_ 6174n/5832



Answers

Use the Green's Theorem area formula (Equation 13$)$ to find the areas of the regions enclosed by the curves in Exercises $21-24$ .
The ellipse $\mathbf{r}(t)=(a \cos t) \mathbf{i}+(b \sin t) \mathbf{j}, \quad 0 \leq t \leq 2 \pi$

And 36 Here here you is equal to X plus Why? And we is ableto why minus minus two. So why indifferent to me? Plus plus two and X is different to you minus three minus minus two So x squared plus two x y plus two y squared minus four minus four y minus eight is equal to is equal toe heel So x plus y squared across Why minus two and squared is equal to this is equal to four plus eight Eso you squared plus B squared is able to it would work so clearly the area is equal to 12 pie

The problem started to lot of buying era on. We can use the former here. So this is a i lips right? A close eye on TV science. So actually a cosigned he no one ever till here to you from 0 to 2 pi and x d y so a Husayn key you I should be be cause i n t e t b and minus y t x So, Linus, why last y t x t x ray the a minus a sign he did right minus a sci fi So just a B right? So Hi, baby is our answer, which is a relic of the lips.

We're going to calculate the area in close by the lips given through the equation X squared about of my X squared plus y squared B squared equals to one and Teoh calculate that we take into account. Um, the equations of the Libs is symmetric Respect to both access means that we can take only this the upper part of the lips which is defined or corresponds to Why were the Americans to Syria? In that case, Theo equation of the lease can be found solving for why in this question So why squared But be square? He's one minus x squared about about a square And so we had that. Why square is he going to be square to better, like a square times a square minus x square And then for what? Positive why we have y equals absolute body Be committed by absolute value If a times a square plus minus Sorry. Okay, times the square root off because we have taken to wearing here a square These were minus X square. And so this equation off the opera part of the leaps and and that is defined for a square minus X squared were then were able to zero, which means that X squares list done. They're going to a square. And so the absolute value of X is less than or going to the absolute value of a. But he's exists in the interval running going through and then the events of the money of a to absolute money of a then the the area enclosed by the lives. He's twice the area the area under the curve given by discretion here and these immigration the this area close with these cure the upper bar, the lease and the eggs axis is giving as the integral for a negative absolute body of a to absolute value obey Which is them in here, Off the equation, absolute value be you went about G square road off a square minus X squared differential of X, which is his question here. And so, uh, this is two times absolutely be better by absolute value, very in general, from negative after nobody obey to absolute value, obey square root of a square minus X square, the French elevates, and this internal here will be calculated using a tuna metrics institution. And for that, we're going to define implicitly a verbal filler through the equation. Mm x equal to absolutely of a sign of theater. And with that began that differential of X equal to it's absolute value, Very cold sign of the difference of theater and that we define for an angle theater in the interval negative by half by have, in order to have an inverse well, they find for these change of verbal and, um, now we can we can calculate the new limits of integration here. That's that. That's it. We're going to change the limit off integration giving this integral. So, um so we're going to say that for for X equal to negative, absolute value, very, which is the lower limits here We had that. A sign of theater is you go to and negative one, and then you have the theater is negative by half and for exit going to absolute value of a, which is the upper limit. Here we have that way have the sign of the reasonable to one so their physical to buy health, then the injured girl from negative to the value of a two absolutely of a off the square root of a square, minus X squared differential of X equal to control from negative by have to buy have of the square root of a square minus he square sign square. I'm theater times, differential of Axis subsidy value of a co sign of theater, three inch Laffita and then a sequel to the up to the body of Ray into a row from negative by half. I have. Oh, he's square root off a square times. One might assign square of the times go sign on theater differential better, and that is equal to here. The square root of a squares absolutely very time sensitive alleyway is a square dural from negative I have to buy have off the square root of on minus sine square of theater times CO sign of the French Are there at least equal to the square into row from negative by. Have to buy have here. We have absolute value of co sign Anthea Time scopes on infinity, French ill fitting, and that is a square times into world for a negative. But I have to buy health off a go sign, square off the differential thin, and this is because there is in the interval negative by half by half and there go sign, if it is possible. And so the secret to pay square integral for negative by halved by half. And these ghosts and score of there is equal to one Blasco sign of two time theater divided by two differential data on Deasy's sequel to a Square integral from negative by halved by half off. Ah, here we have to because we can get it out of the integral. On Inside we get along, plus co sign of two time theater, Fragile theater, and that is a square better by two internal from and here we were going to ride the primitive of thes function here, which is theater bless sign of 2/10 data. You better be too. And that civil waiting between developed the points negative by half and by half, and these equal to a square do anyway to at their equals by half by half, plus sign off by the by the way too minus at negative. By have we have negative. If I have minus sign off near defy everybody way too on these equal to a square to whatever two times, and we know that this is zero This is zero and we have I have a blast. I have That is, by hey square by better by two. And so remember we said at the area enclosed with the Libs is equal to here two times, Absolutely be there with absolutely of eight times the Internet we have just calculated. So you can say that the area come close but he leaves a sequel to the times. Absolute value be given in absolutely very times. These in true we calculate here is a square by health. So we have to simply five with this two and a square with absolutely a very we know that is equal to absolutely be times everybody of a times by these two body of a times absolute body be terms by And this is the area enclosed by the Libs.

We're supposed to find the area off the two dimensional region that's given by this equation here, and many a few might have seen this before. It's called the equation of an Ellipse and responsive deduced area for Iowa. Now here there is no assumption on the science of A and B, so we have to be careful about that. And also there there are easier ways to find the area issues of Dublin through all. But since we're on the topic of single cackle ish calculus of single variable, let's try to stick to that. So the usual method here should be trying to isolate wife acts as a function X. So from here, if you do this, we're going to find this is plus minus B squared and one minus X squared class last a swear. And so this is again plus minus. Absolute value of being comes out of the skirt because we don't know whether he's positive or negative and the same thing here. So this is a square root of a quadratic function, which, in general we don't have to deal with when the integral sarin boss. So So how do we find the area Well, if he feels sketch to do a little sketch here wax X axis, it's going to be wealthy. Top portion. Well, so So if you look at the balance for X, we're gonna have X is between absolute value of a and native outside value off, eh? Precisely. Because we have this term inside the skirt and we want that to be a positive, not negative. So let's say this is here, and this is here and then there. And then the graph looks something like elliptical shape that looks like this I'll say will go like it will go like that. Here, sin a symmetric. Wait. So here's what makes sense here. And one loss a simple fellow work is if we only find the area off this part and the multiple right by two, and they will give our oh answer. So its focus on this area first Well, which means we're only look at the function. Why've sacks, of course, only deposit of squared of our formula. So now we have a well defined functional why, in terms ofthe in terms of two independent variable acts, and we know that the area here is off dysfunction is going to be simply the integral From the end points over, Dan points into role and my ax the ex here. And then after after all of this, we're gonna have to want to buy this way to to get the area or we can white simply that is actually two times this. And this is our actual Astor Final answer. I think that's a better idea. So here, if you write this out in a little more detail, is going to be two times native. Absolutely. I've a absolute life, eh? Square root off one minus x squared a square d x here. Now also, we are on the chapter of tricks substitution. So let's try to do that way. So the simplified a square it we the whole This whole thing here suggests that we should we do the substitution. It looks like this accent because I'll survive a Times Co sign their sign. Either one of two more will work, really? So it's just pick sign here. Sign of tea. Well, now we know that X is going between negative squirt of a negative. Absolutely. If a and prospects of positive outserve value of a so this means we have sign It is going between NATO one and positive one. And this means we have the variable t. Well, our Pamela or tea will be going from native pie over to positive pie over to here. And so and the final thing to do is we need to figure out the integration of our it seems that it seems I have for gotten the squared of be here absolutely of b here. So coming back D X is equal to absolute value of a I'm co sign of X time's DX and this our function instead of square tonight is also going to be co sign. Not that I, um actually not ex hold very wasn t t 18. Finally, the expression inside the square there's squared off four minutes x squared. The right by a squared is going to be simply course I nifty note that I commited the square absolute values sign outside of this course sign because in this range in the range, that's in the interval. Negative pirate to enforce. If I were to core science always known ater so that makes our job a little simpler. So continuing in this fashion we will write that our area are actually final. Answer is single to two times now it's a trick control eight of pirate, too. I over to and let's see we have I have constant out survive be here and that survive a here and also have Cose I Inti one course I Auntie and two courtside of tea. So this whole joint this is going to become out survive eight times out Survived The Times Co sign Squared off t me, too. Now this is a rather simple Trigon through and we can use the double angle. Course I inform relate me different, so we'll take the two constants out. And this will become native pirate, too. Or is there a fire, too? And one plus car sign off to team you two and we can write down the answer immediately now, So this is going to be AB survive eight times out of o B. Times war integral off one from pirate to two native pyre itude That's just pie blanked off the interval, plus same course. Same constant times. Well, in theory, off coast of two t, there's sign off two to the right of way too valued at the end points. What do you see? That Dan points, give the value sign of pie and sign off, native by both of which are zero. So to finally answer our area, turn out to be pi times. I've survived eight times and survive bee note that have this generalizes thie formula for circles when, if nbr so if if a If they have the same line into that which we will call our and this gives dairy off a circle that's pi r squared. In other words, a circle is just a special case off ellipses.


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