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Using polar coordinates, calculate the double integral fI sin( Vr2 + y2) dxdy where D = {(€,y) 22 +y? < 1}....

Question

Using polar coordinates, calculate the double integral fI sin( Vr2 + y2) dxdy where D = {(€,y) 22 +y? < 1}.

Using polar coordinates, calculate the double integral fI sin( Vr2 + y2) dxdy where D = {(€,y) 22 +y? < 1}.



Answers

Use polar coordinates to evaluate the double integral. $\iint_{R} y d A,$ where $R$ is bounded by $r=2-\cos \theta$

In this problem. We're trying to find the integral of X d a over the region r and R's the region bounded by our small r equals one minus sine theta. So here in the red, I've drawn a picture of one minus synthetic in rectangular coordinates. And now I'm going to convert to polar coordinates. When the angle is zero, the radius is one. So that's here. When the angle is piper tooth, the radius zero. When the angle is pie, the radius is one. When the angle is three pi over to the radius is two and then back to zero again in a positive way in a positive way. There you go. So it doesn't have any holes in it, any loops or anything like that. So we're just gonna integrate regular? Yeah. Okay, so X and polar is r cosine theta and d a is R D r d theta and our is going from zero out to that curve. So 0 to 1 minus sine theta. And then to get that whole picture, I had to do it from 0 to 2 pi. So 0 to 2 pi here she so I have r squared ups 0 to 2 pi integral of r squared R cubed over three from 0 to 1 minus sine theta. We still have this coastline here because I'd say had a deep data. No. So now plug in the inter grams the limits of integration 0 to 2 pi one minus sign cubed. Let's put the one third out in the front so I don't get confused with it. All right. The easiest thing to do here is let you be one minus thesis ein of data. Then do you will be minus the coastline of data data data. So we have the coastline theater. We have the d Thetis. We need a minus sign in here, so I'll just put a minus. Sign out the front to take care of it. Alright. When Fada is zero, then you is one. And when data is two pi, you is one. So the next dinner girl is gonna be negative. One third, 1 to 1. You cube D'You, which we know is zero. Because you're integrating from 1 to 1

So we wantto right this integral and pull a trick ordinance and then evaluated. Okay, so the first thing that we wanted to know is that this is equal. You are squared. So our square is less than or equal to floor less than able tto one. So are ours. We're going to range between one and two. All right, so we've got basically these two circles concentric circles. One has a radius of one one has a radius of four. And our wives we're going toe range like this. Right? This in red. So why is equal to X? And the lower bound is wise, equal to zero. Good. So we've got this region here as regent or hey, and we can see that our region are yields The fact that they does going to run from zero to four pi. Okay, So theorems from zero for a box. She Excuse me, pirate, for our runs from zero to one. But sorry. I wanted to You got arc ten. Now we have to write why, in terms of a polar in full accordance and they know that wise, even to our sign, data and X is equal at Arcosanti you need this? Diar? Artie! Artie! All right, So what happens? These guys cancel out in this term right here is really just tangent data. Good. So does your toe. A higher floor. One to two of our mark can can data d'Or be there now. Looking at mark ten of tan of data. We will get justice first. Coordinate. We know that this is just going to be equal to data. Always So are integral Would really simplify all the way down is our time's data gr data. Not this other interval here. All right, so let's actually do this. Integration. You've got R squared over two from one to t data data data. So Wade got injured girl from zero to fire before two minus one half. Um, but the fate of potato Okay, so we really got three over too. Times data squared over two from zero pi over four. So is that three over four times high? Well, er, floor squared. Okay, then this will give us our answer three by squared over sixty four

I think so. In this problem, we're going to be solving the equation, which we have a d a. So that most likely normally sense for area. And in this case, we have this area bound in the first coordinate, and it's above boundary is this circle, and it's boundary from below. Is this line so we can draw this out just to see what this was exactly look like? So this circle this a bead two. We have one. It's gonna be like this on our line is, like, said a bit more like that. Um, so it's this region. Okay, now from here, we're gonna do is we need Teoh first. I'm gonna write out, actually, um, three helpful questions so X is equal to our co sign data. Why, it's equal to our science data and R squared is equal to exported plus y squared. So first thing you can do is start trying to solve for r R. Um, and we're going to take our equation up there. The x minus one squared plus why squared is equal to one, and we're going to start plugging in the values for X and for why? So we have our coasts. It Ah, minus one. All this squared plus are stop sign data. All that squared is equal to one. This isn't the r squared coasts squared minus to our coasts. They, uh, plus one plus r squared sine squared data is equal to one so we can subtract are once and we get rid of that. And so when I say this is equal to zero, but just bringing the don't go this way. Um, so we're gonna have our squared coast signing squared, beta plus r squared, sine squared, beta minus two, our coast. I'm gonna move that last part. The other side's We're gonna add that both sides So what? Ever Plus it here. So that's going to give us to our close. They equal to r squared, Closest squared plus R squared sine squared bird. Okay. And now from here, what we can dio is divide out and are with my daughter. Arm would have to Coast data is equal to our coasts. Square data plus are signed squared data. And now we can take out our are gonna have coasts. But the plus sign squared data, and this is just equal to one So we're left with coast to coast. Data is equal toe are so that's one of our values for our right there. Andi Now, from here we're going to do is, uh I'm working out actually kind of solve for our data. So we have X is equal to y and R X is equal to our coasts data and our why is equal toe our sign data so we can divide. Um Why there are and cross out are ours and we're left with Kosaka and data is equal to sign Vada. And we know that that is true when Ada is equal to hide over for in the first coordinate. And so we have, um right now, two values. Um and what we're also going to do is because we're in the first coordinate. We're going to say that our fada is equal. Teoh hi. Over to the information that we have have been equal to hi over to It's in the first quarter and it's in the proportion First coordinate and we're going to have our r is equal to zero. And now we need to take our original equation up here and we're going to have to put that in polar coordinates that we have to. Why? And why is R sign fate? Us. That's all we need. And now we have our creation that we can put into a double integral with all of our polar values. So we're going to have from hi over or too high over to and for zero to two coasts sita of to our sign data. And then we have our d a, which is just going to be R D r de data. And now we're just going to separate our parts that have oh, are on with them in brackets and can start solving are into girl. So we're gonna have to r squared t r. And we'll have our sign data over here. Okay, And now the integral of Are squared is equal to are cute, so it's going to be our cubed over three. So we're gonna have to our cubed over three from 0 to 2 coasts Sita on this is times sign Data de data and we have the integral from however, four to I over to, and now we can start putting in our values. So from pi over two of and we're gonna take our to, um, Earth's out to make this easier. So of our cube. So in this case, we have to co sign data cubed minus zero cubed. This is a B times sign 70 data. This city is equal to zero. So that is going to get crossed out. So we're going to now have our Andi. We can cube too. So two cubed with eight that we're gonna have co sign cubed sign data. BP data on. We could take out our 8/3 on driver for two. High over to. And now from here, we're going to do IHS, um, we're gonna have co sign as you we're going to do a u substitution with Our co sons were going to say when you do it over here, So you is equal to co sign data. So be you is equal to negative Stein data, be it Seita. So do you. Over a negative sign. Data is equal to D data and I are gonna plug in. So we're gonna have Are you cute? Sign Saito. Times are e u. Negative sign. Data you can cross out are in betas and we can take our negative to the on. We can also start now solving for our introduced. The integral of you Cube is you to the 4/4 from pi Over four to I over to and are you is equal to host Sign faded in case you forgot. Now we're just gonna plug that value. And also, this is all times. Um, so we have negative three times co sign before data over four minus Well, not minus her. Um, we have a collection of values. It from I report to you pi over two, and now we're gonna plug in her value. So wait, three kinds coastline for it. I dio over minus coast line fourth, I over for over, for so co sign of pi over two is equal to zero. So this is just all gonna be with cirrhosis is gonna cross out, and so we'll have negative 16/3 from negative. Um, on. So the co sign of pi over four is equal to the square root of to over two. That's going to be for over for on DSO we're gonna have this is equal to 16 over 12 times. Um So the pot square root of to the four is going to be equal to for on our denominator is going to be equal to 16 cause too four is 16 and so we're going to have that this is equal to one third on that is our answer.

All right. So you want to solve this double water? Go over our two x Y d a. Where are is a region where are goes from 1 to 3 and data goes from zero to pi over two. So we have to use our following convergence to polar coordinates where X is Arcosanti, Dana. And why use our science data and D A is just already already data. Now we can go ahead and start solving. Ah, we can fully substitute this with our bounds with our bounds. And then, um sorry. I should be a one 123 of two times are co sign data are scientist org already there, which is just our to our cubes. Science data co, science data. Do you already data? Now we separate these intervals. We end up with the following, and then we can just, um, go ahead and plug it all in. So we, um, with this first integral well, noticeable have signed data and then it's derivative cosign data data, a za product. So then we can just This is basically in a role of you, Do you where use equal to sign data. So this ends up being two times signs where data over to do you own a pirate too. And the second interval just becomes our to the fourth over four from 1 to 3, I pi over two Sinus one and then zero sign of zero. So this ends up being two times 1/2 and then this integral, we get three to the fourth over, four minus 1/4. So these ah, ones in 1/2 and two cancels. Then we're just left with 81 minus one all over four or 80/4, which is just 20. So there's integral value. It's the 20 in


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