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CacotonEis the solid below the paraboloid z = x2 + V2 and above disk x2 + v? < 4.a) Sketch the solid.b) Using a double integral in polar coordinates, find the vo...

Question

CacotonEis the solid below the paraboloid z = x2 + V2 and above disk x2 + v? < 4.a) Sketch the solid.b) Using a double integral in polar coordinates, find the volume of E:

Cacoton Eis the solid below the paraboloid z = x2 + V2 and above disk x2 + v? < 4. a) Sketch the solid. b) Using a double integral in polar coordinates, find the volume of E:



Answers

Use polar coordinates to find the volume of the given solid.

Under the paraboloid $ z = x^2 + y^2 $ and above the disk $ x^2 + y^2 \le 25 $

Okay, So this problem wants you to evaluate the triple integral of exposed wide Losey over the region off E. Where is is a solid that is, in the first Occident and is below the pair up. And it is below the probable oId Z equals four minus x squared, plus y the evil four minus x four miles y squared. All right, so the first thing we do is to convert everything into cylindrical coordinates. So if we were to convert our into grand into political coordinates, we would have extra swipe a Z where X is our co sign down our coastline. Dana, we have Why is our side data our scientist? No. And we have the just the and D V in cylindrical cornices are times are times DZ, pr di fada. All right, so now let's look at the boundaries. Yeah, in the first. Often that means our X y and Z components are all positive. Traveling really mattering cornets and our problem oId z before my eyes expire, Biswas, where will that can be rewritten as e equals two or minus r squared? And this is a downward facing problem. So looking at our boundaries versus DZ. So the bottom of our Z is the X Y is the X Y plane because I remember one of our arm. It is in the first octave. So x Y plane is The downward plane is a plane of Z equals zero Essentially, that restricts the region. Then we have the downward facing parable oId Ziegel. Four mice are squared, which is the upward restriction. So we could just let in before my eyes are square. Yes, we have the are So er and this obviously this largest one c equals zero because is done, we're basing problem. So we just said 0 to 4 minus r squared we can get our off to So our radius goes from 0 to 2 and finally our angle theta d'hara. It goes from zero two pi house. Now why is it pie house? Well, it's pie has because is in the first oxygen. So when the X y plane, the X and Y components have to be positive. All right, so now that we've got everything is little coordinates and we've got our boundaries now it's time to calculate it. Okay, so let's first integrate this with respect to z So our boundaries or the R and D d and I say the same. So if we integrated with respect Dizzy Well, I knew it. First get R squared the equal sign data our squares equals and data plus our square and the science data whispers he signed via was 1/2 rz sward 1/2 our a z squared And the boundaries are, of course, from four minus R squared from 0 to 4 minus are square and the other two or d r d Fate up. Okay, Next if we plot If we were to plug in the points for Mayes R squared and zero into our into ground would get the double into grown off double integral of R squared times four minus R squared. Course I do. You know waas R squared. I was four minus r squared. Sorry, they are waas. Well, it's 1/2 are times four minus R squared squared DRD data. Okay, so now we have to integrate. Now we have to integrate this with respect to D I or d thing up. Well, looking at this equation, you can probably tell that this would be This will be a struggle to integrate with respect to our So how about we try integrating it with respect to defeat data first? So let's swap the boundaries to detail. They are because if we did, with respect to D data and, er we're bound to get since we have a co sign and a sign when we're dealing with a region boundary a pirate, too. We're about to get a zero somewhere, so we'll probably end up most nicer than if we just did this ugly Integral will just back to our So if we were to integrate this with respect to data would have an integral from 0 to 2 to the boundaries of our we would get into grown r squared. I was four miners r squared. I'm silent. Data minus R squared times four minus R squared coastline data plus 1/2 one house are times for monies. R squared squared times stayed up and this is from pi over two Priore from zero to Pi Overton Oh, respected data with respect to a d. R. So now if we were to plug in our boundaries, uh, from zero department tomb, we would end up with an integral in the winter grown off of our square Four minutes or square four miners are swear plus pi over four times are times four minus R squared, squared, plus are square All right, four minus R squared, squared because off the coastline so would be at this. Oh, yeah, and respect the d r When were you just being combined? These two terms. And so we were to combine those term we could get a simpler into grow from 2 to 00 to our square I was four minus are square plus pi Ford's Our times four minus R squared, squared Respect the d r. All right. And if we were to expand this out so we can further simple fire we get an integral you go from two from zero to off Oh, yeah and be distributed to r squared into parenthesis east into our before my eyes are square Viet eight R squared When it's to our to the fourth to our to fourth plus pi over four our times 16 are minus eight R Q plus our bit to the fifth power Do you are? And now that we have this, we can integrate with respect to our to get to get in. Over three are cute. Minus to are to the fifth over. Five plus hi over four times IMEs times eight 18 r Squared. When is to argue before? Plus our six over six from 2 to 0. Okay. Wait. I see my hair here. This are not supposed to be there. I already distributed the are. So if now if we're simply plug in the values from zero to, we get a final answer of woman 28 over 15 plus a pi over three. And this is your final answer.

So, um, oftentimes we can't just take a single integral. We often have to take double integral Zoran this case triple integral in order to find important values. In this case, we're finding volume. We also know that it's important to be able to switch between coordinates because, um, things aren't always rectangular. Oftentimes it's easier to use cylindrical or spherical coordinates. Eso What we're going to be doing is working on our ability to use those different coordinates to make problems ultimately easier. So what we have here is we first want to determine the range for theta. Um, since we know the regions in the first Occident, the angle will range from zero to pi over two. Then the Z value. We want to convert into rectangular coordinates. Since Z is equal to four minus x squared plus y squared, we will just call this four minus r squared since x minus X squared minus y plus y squared is R squared. Then we know that when Z uh, we can plug in Z equals zero. When we do that, we see that our is equal. Teoh a radius of two. So with all this in mind, we now can take are integral. We know that the fate of value is from zero to pi over two. We know that our radius is from 0 to 2 and then we have our functions E which is represented by four minus r squared. And therefore we have our X plus y plus z Easy. De are de Seda. Um what we can dio with this is, um, move the theta right here. Move the theta, right? Yeah, When we do that, we know that this is just gonna be pi over two. Similarly, we can we want to represent And before we do that actually will want to represent X, y and Z as the appropriate, um, variables. So X we know is going to be our coastline. Fada, Why will be our signed Fada and Z will just be easy, but we want to have it's going to be our DZ drd fada. So when we multiply everything through, we get our Z r squared sine data R squared coastline data evaluate this simplify, But we can also use these calculating tools because they're very useful in finding inner grows quickly s So what? We end up getting is this? And another way we could write it if we salt it out, would be 128 over 15. Class 8/3. Hi. Um, as we see these answers, check out. So we know that we did it correctly, and this is how we will conclude this problem.

So, um, co sign of zero is just going to be equal to, I mean, co side of status some people to zero when data is equal to pi over two on day three. High over to. So we'll have are high over to and are 3/2 and that is it. That is our answer.

Hi. So this problem we have to find the equation for what the volume would be in terms of double integral using information we have. So, um, we have two equations right now. We have one. Um So the volume comes from inside this Andi out side of this equation. So the first thing we have to do is we're going to try and find our our values. And there are three equations that are really helpful when doing this type of work. Ah, where we're switching from, um, the coordinate plain into a polar coordinates type of a ah, situation. So we have excess equal to our coasts. Fada, why is equal to our sign? Data and R is equal to x squared plus y squared. And we currently have premised two equations that look exactly like the last equation I just wrote. So, um, her 1st 1 we have X squared plus y squared is equal to one. So I'm sorry. And this is our squared. Um, so are is just going to be equal to the square root of one which is equal to one so ours equal toe one. But I'm actually write that it different color. Um, so are is equal to one. Let's check with our is equal to one Onda. Um, now taking our other equation. We have X squared plus y squared plus Z squared is equal to nine. Um, and this would be actually like a skiers. But if we're looking at it as if it was a circle, Um, and we just said that Z was equal to zero on. We kept the same equation, we would have the square root of nine, which is equal toe three, um, is ableto are so r is equal to three. So we have two values now for our So we have our is equal to one and three. Andi, we have to now find what our values for our data part of our equation is. So, um, we're going to just use our first equation that we solved. So we have, um, x squared. Plus why Squared is equal toe one and union for our values squared. Plus are times sign squared is equal to one. And so what we can do is just they are going to be one, because we know that it's one for this one. So we have co sign squared data plus Stein squared data is equal toe one. So from here, all we have to do is we'll say sine squared theater is equal to one minus co sine squared data, and all we have to do is figure out when ah sine squared data is equal to one minus co sign square data. So there, two times that this would be true and that would be so when sign is equal to zero. Ah, that's why Sorry. When they is equal to zero sign is equal toe zero. Um, I would have one minus and then co sign is equal to one when fate is equal to zero. So this would be one for one co sine squared, and that's equal to zero. And then the other time would be when Sinus equal to high over to when sign is equal When time is it with the pi over to this is equal to one so sine squared. Uh, this just be one. Still, this would be equal to one minus so co sign when, uh, data is equal to pi over two is zero. So this is just going to be zero. So those are the two cases or datas are gonna be when, um zero and high over to okay. And now from here, we're going to take our, uh third. The third are sorry. A second equation that we had, which was, um X squared plus y squared plus Z squared is equal to nine and we're going to solve for r Z squared. So we have C squared equal to nine, minus x squared minus Why squared Onda? We can just rewrite this as C squared is equal to nine minus X squared plus y squared. And we know our expert plus Y squared from our equation up here is equal to R squared, So this is equal to r squared on. So if we're solving for C, we're gonna have to take this. We're route of both sides, so I just take this route and so we're gonna have Z is equal to nine minus are squared, and from here, we have enough to form our equation. So our equation is going to be for our fate of values. We have zero and pie over two. So we have zero pi over two. And for our our values, we had one and three. So from one, 23 Andi, we're gonna have here. What are zebras, which is nine minus are squared and we're gonna multiply that by are our that we normally have. And then we have our d r de vita, and that is our equation.


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