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(10 points) Convert the following double integral from rectangular coordinates to polar co ordinates Then evaluate it.dy drHint: remember the half angle formulas:c...

Question

(10 points) Convert the following double integral from rectangular coordinates to polar co ordinates Then evaluate it.dy drHint: remember the half angle formulas:cos" (8) = % + % cos(20)sin?(0) = % cos(20)

(10 points) Convert the following double integral from rectangular coordinates to polar co ordinates Then evaluate it. dy dr Hint: remember the half angle formulas: cos" (8) = % + % cos(20) sin?(0) = % cos(20)



Answers

Evaluate the following integrals using polar coordinates. Assume $(r, \theta)$ are polar coordinates. A sketch is helpful. $$\iint_{R} 2 x y d A ; R=\left\{(x, y): x^{2}+y^{2} \leq 9, y \geq 0\right\}$$

I'm going to integrate this function. Why? Over the area are where are is bounded by our little r equals two minus coastline data. So first, I'm gonna draw this because I need to find out if it has any holes or things to worry about. So first I'm gonna draw in rectangular. So minus the coastline of data would look like this. Go through here. Here, here, here and here. And so then if we add to to it, it will go to to To at 22 Looks all right. Eso then in polar at zero. I am at radius one at Pi over two. I'm at radius to at pie. I'm at Radius three. Uh, 35 or two. I'm out radius, too. And then back to zero again. So here, here, here. Here. It's sort of a little cir cle thing like there no holes are loops or anything like that. Okay, so we're gonna integrate using polar coordinates so you can see that data goes from 0 to 2 pi because we're filling it with little thing. Little pieces of area like this. Our goes from zero to our, which is two minus cosine theta we're integrating. Why? Which is our sign Data. And then Dia is r d r D data. Okay, so is your Did you buy? You're two to minus cosine theta We have an r and and are so r squared sine theta Do you already stated? Alright, The other girl of r squared is r cubed over three So we have 0 to 2 pi are cute over three from 0 to 2 minus cosine theta sine theta d theta. So one third 0 to 2 pi tu minus the coastline of data cube sine theta d theta. We don't need toe Cube that up because we can call call either to minus the coastline data you or just a coastline date of you. So that's what I'm going to dio I'm gonna let you be the coastline of data. So then d u is negative Sign data d theta. And when they did a zero than you is the co sign of zero, which is one. And when the data is two pi than you is the cosine of two pi, which is also one. So in the next step, I'm gonna have one third integral 1 to 1 Tu minus you, Cube, but needed on my sign in their a minus Sign out there, do you? So I'm integrating from 1 to 1 So the and a girl's gonna be zero, so I didn't really have to integrate it at all.

Let's go ahead and sell this double. What role of over are heated? Negative X squared minus y squared D A where we have all our is just reading extra plus y squared is not very good tonight. So we know that this region just the radio a circle of radius three. Where in polar coordinates This should just be ours. Lesson record three And they ah is here to two pi. So we'll end up with found The interval of this part is just negative r squared. We have e to the negative r squared Hardy already data separating our inter girls. We end up with this giving us a two pi out front and ah, are you the negative r squared? We go ahead and integrate. All right, you to the negative r squared d r gives us use of is the negative r squared. Do you is native to you, Do you? It's been end up with negative 1/2. I think all of you to you. Do you giving us? They would have. It's the negative r squared. So we'll have night of 1/2 you too. Negative R squared from 0 to 3, leaving us with the following, UM, two pi times 1/2 e to the zero minus e to the negative nine. So we end up with just pi times one minus negative. Ninth as our answer.

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

Turn and solve this very simple polar girl. And I mean integral one called accordance. We know that this is our region described problem. And, um, we know a simple substitution that this is just r squared and pulled. Afford it. So then reading this all at once being the following and, um, separating articles. We have a pretty simple job here. Hand. So 60 day, That's all right. So we have, um two pi is our first inter goal. Let me get are to the fourth over 404 And, um yeah, so we end up with forward to the fourth number four, which is for Cuba or 64. We're gonna go integrates to 128 pi.


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