Pcin} The elipse below has a major axis oflength &.2 cm and a minor axis oflenguh 5.4 cm (both are showr as dash lines) Find Ihe area ofthe shaded region ifthe...

Use a line integral on the boundary to find the area of the following regions. A region bounded by an ellipse with semimajor and semiminor axes of length 6 and $4,$ respectively.

In this problem we are given to polar curves are equals one plus go scientist to and R equals three co scientist to I'm were required to find out the region That is inside the curve r equals one plus go sign of data. An outside R equals three co sine of theta. So we'll use the formula for the area in polar coordinates. Rather the double integral formula that is the inter girl over the region R. Of our D. R. D. Theater. No a region are is are going from so are goes from Mhm. The region that is inside to the region that is outside. So that's one plus go sign of data. You three go sign of theater and our tita Goes from the first point of intersection. It's called it eras of one for a second point of intersection that has cleared us up to but for that we need to find the points of intersection. So we can equate these two equations so that's one plus cosine of theta. That's right, okay. and three co sin of theta and we can solve this equation. So that's bring it down and try to solve it. No we can transposed one of the co sin ti does to the other side. So that's one minus sine of theta. That gives us one equals to go sign of tita. And that gives mhm go sine of theta Is equal to 1/2. Now the co sign is positive so are required region must be either in the first quadrant Or in the 4th quadrant and we can find the the terminal ankle Let's call it alpha Using the formula the in risk co sign of 1/2. And we can use a calculator To evaluate the in risk co sign of .5. And that will give us bye bye. Three ratings. No. And in the first quadrant our angle is simply equal to the terminal angle. So our first answer Status of one equals bye bye. Three nine The 4th Quadrant. Yeah. Start second point of intersection. In the fourth quadrant it's given by two by minus alpha. And that is Dubai minus bye bye three. And that will give us five by by three as the second point of intersection. So our region goes from Tita from by by three 25 x x three. Yeah. And we can take our integral. So that's a equals. So the limits of integration of to go from bye bye three 25 x x three. The limits of integration of our go from. Yeah the inside leaking to the outside region. So that's one plus the co sign of data. Okay. Mhm. Yeah. Do three times the co sign of data. Yeah. And the inter girl is our time's D R. And D theta. Now let's consider the inner integral. So that's Integral. one. Blasco Sine of Theta. 2 3 times the cosine of Theta. And that goes from rather it's the integral of RDR. So this evaluates to and simply use the our rule for integration. So that gives us our squared divided by two. Going from one fiscal sine of theta trico, Sine of theta and that gives us one plus Yeah That is one plus Go sign of Tay to the whole squared divided by two minus three. Co Sign of Theatre. Okay Rather I exchanged the order to be the other way around so that's three go sign potato the whole squared. Okay now we can expand this and that becomes nine times go sine squared of theta divided by two minus one. Blessed to co sign data plus go sine squared of today to divided by two. Yeah yeah and Emerging these fractions will give us nine times the cosine squared of data minus one minus to go sign of data minus that. Go signs squared of data. The whole divided by two. Yeah and that simplifies to eight times the cosine squared of theta minus one -2. Go sine of theta divided by two. Now this will again need to be integrated and it's not easy to integrate. Cosine squared if any angle teeter. So what we do is we use the identity. So cosine squared Theta is equal to one. Bless Go sign of duty to divided by two. Identity. We're gonna go ahead and substitute that in so that's one plus go sign of duty to divided by two minus one -2 times the go sign of data. The whole thing divided by two Can reduce these this just becomes four. That gives us four plus for co sign of do kita -1 -2. co sign of data divided by True. Yeah. And the soul just becomes For co sign of two Theta. Yeah plus three minus two times The go sign of data divided by two. And we can reduce all of this by two and that becomes two times go sign of data duty to It was three x 2 -2 times the go sign of data. Now we can go ahead and substitute all of this instead of our first integral. So that just becomes Peter going from five x 3 five x x three. The integral of do dimes go sign of do tha tha It was three x 2 -2 times the go sign of data. Dames de Tito. Yeah. Now the integral of co sign is just sign of data. So are integral becomes two times sine of do theater. Yeah. Mm hmm. Divided by right. The derivative of the angle. So do the derivative of two. Theta is simply too. So we're going to divide that by two. The integral of three x 2 is just three x 2. Data. With respect to data. Thank you. Mm So that's three x 2. Data -2 times the sine of data. Going from by by three, 2, 5, 5 x three. And this entire thing turns out to be two times sine of two times 5. 5 x three Plus three x 2 Times 5. 5 x three minus two times. Yeah. Sign of 5. 5 x three. Yeah, yeah minus two times Sine of two times. Bye bye Three. Really? That's three x 2 Times. by by three oh minus. Do sign. Yeah. Bye bye three. And we can evaluate this entire expression using a calculator. And this turns out to be all right, 8.02. So our area is 8.02 units squared from and that is are required answer using a double integral.

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.

So if a problem for before we have to find and sketch the area between the given pounds. So first, let's try toe sketch the area So we have our X and Y axes and the functions literally just a spiral. So if we plug in zero is just gonna be a zero So here, if we put in pi over two, it's gonna be pi. So it's gonna be somewhere up here And if it plugging chi over to If I so somewhere here and Novia spiral tonight this and the areas of going to be bounded between this viral and the X axis so spaces could leave this bridge on here. And with that, don't let's go ahead and try to find the area. So the integral we're going to be using for these kinds of problems 1/2 times the integral from al fits of Gaeta of the Function Square and the data in our case, the functions and literally just to say that. And we already have our third of balances from zero supply, so we just plug them in. So if you anyone have times the ends a girl from 0 to 2 pi of two fatal off. That's guarantee theater. If we square those out, it's gonna be fourth in scoring. And if you move the constant multiple four to the outside, it's going to give us two times the integral from zero to a pie. Say that scored the data. And if we integrate its just give us 1/3 time, stay that cubed and this from zero supply. And if it plugging values for Justin get five killed over three minus zero. And since we have a constant multiple of sue in the front, we're going to multiply our results by two. So in the end, our final solution will be certified. Here's all over three and, yeah, that's basically adds.

Problem number 34 uh, area is equal to integration from one the three open two over Theatre Square deflator, which is equal to two integration off 1/3 to square deflator, which is equal to two data for negative one over a negative one. From 1 to 3, she is equal to negative two ah three for negative one minus minus one, which is equal before over three.