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0/1 pelnte Prevkous Anbware sCakETd 15.7.029.HolAnk Your TeacherEvaluate the Integral bv changing cylindrical coordlnates 81 _ 72 [J dz d> 81 -v Vx +v2Need Help?...

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0/1 pelnte Prevkous Anbware sCakETd 15.7.029.HolAnk Your TeacherEvaluate the Integral bv changing cylindrical coordlnates 81 _ 72 [J dz d> 81 -v Vx +v2Need [email protected] 0 Iale]Submit Answer Save Pronns>Praclce Anobher Version

0/1 pelnte Prevkous Anbware sCakETd 15.7.029. Hol Ank Your Teacher Evaluate the Integral bv changing cylindrical coordlnates 81 _ 72 [J dz d> 81 -v Vx +v2 Need Help? clkaull [email protected] 0 Iale] Submit Answer Save Pronns> Praclce Anobher Version



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$29-30$ Evaluate the integral by changing to cylindrical
coordinates.
$$\int_{-2}^{2} \int_{-\sqrt{4-y^{2}}}^{\sqrt{4-y^{1}}} \int_{\sqrt{x^{2}+y^{2}}}^{2} x z d z d x d y$$

All right. So you want to go ahead and solve the Portugal already of X plus Y plus E. T. V. Yeah but it is just uh the region in the first act in yeah, bounded by the probable. Oid Z is equal to four minus X squared minus Y squared. Mhm. Now we know that the first accident Gives us two bounds on our payroll. So we know that it means that Z. is greater than zero and that data must be between Pi over two and 0. And to find your balance and arm we can just said um are tabloid equation which in cylindrical coordinates is for minus R squared Equal to zero. Since we know that's are found. That's the x. Um the bottom surface that we're talking about. So this just gives us R squared equal four or articles to. So then our last um val dishes that ours between zero and two. So you can go ahead and convert this entire thing. The cylindrical coordinates knowing that devious shifts are easy t already data. So this just ends up giving us It's a girl from 0 to Pi I'm sorry not to brian pi over two. It's a girl from it's here to to to go from 0 to 4 minus R. Squared of our coastline data. Which is X. In cylindrical coordinates R. Plus are assigned data. Which is why. And let's see our dizzy. Do you already paid up? Yeah let's go ahead and drag this interview down here and we can go ahead and multiply it out with ST bounds giving us our square times co sign data. Scientific data plus a Harzi T Z. D. Already data. Right? So the first thing you can do is integrate with respect to Z. Okay. Mhm. Which should give us since this does not have a Z. And this does we can just go ahead and plug in our bounds immediately. So we know that this term will just live. Leave us with um Z in front and then this term we'll just leave us with as you squared over two. So if we go ahead and plug in our bounds we'll see that. Um This first term becomes R squared co signed data for science data. Mhm. Times four minus R squared and then zero gives us zero. So you have to worry about that. And then the second term gives us our times four minus R squared squared all over two. And then again 00 So we don't have to worry about that. Do you already data? Mhm. And then we can go ahead and simplify this further. We can go ahead and multiply both out. So leaving us with are you sorry for R squared minus hearts? The fourth could decide data. Side data. Okay. Plus and then we're gonna go ahead and square this in it. Let's apply in the yard at the same time. So we have All over two. So this gives us um 16. We'll go ahead and do the R. After. So we have 16 -8 R Squared Plus. Hard to the 4th. And then if you go ahead and put the the So the R. N. We have 16. Our last eight are acute. It's hard to the fifth. The arctic data. So now we can just integrate respect to our giving us the following. So we have 4/3 R cubed minus hearts to fit over five times. Co science data. Science data. Oh and then we can do this on our hands. So this this term becomes a it R. Which will give us for R squared. This term becomes a four hour cube which gives us minus are to the fourth and this term gives us Artists 6/10 or 12 from Jared to do data. Now we can go ahead and plug in our bounds. We'll see that zero gives us zero for all the ours. So we can go ahead and just see that this is four times R cubed. So 24 times 8/3 32/3 -32/5. Because I'm fada was signed data plus We have 16 -16 plus 64/12. You pay them This cancels 64/12. I just equal to 16 over um three. So then we have It's a girl from 0 to Pi retune 32/30 minus 32 or five and go ahead and do that on the side. So we have 32 times 1 3rd 25th Which is equal to 32 times a five -3/15 Or 64/15. So we have 64/15. Co science data science data plus 8/3 are sorry 16/3 you data. And then we can go ahead and integrate this. Finally giving us 64 15 scientific data minus coastline data Plus 16/3. They don't From 0 to Pi over two. Okay. And then we can go ahead and evaluate this 64 or 15 times will separate both. So this is the left parts. We have sine of pi over two which is one minus cosine of power to zero and then subtracting sign of zero which is zero. And then subtracting my minus cosign theta, tau and Kazan zeros once we have this and then plus 16/3 times prior to This is just one plus one or two. So we have 128 over 15 to us eight thirds five as our final answer

For giving an integral. We were asked to evaluate Simple and the role of X plus y plus C off the region K, where he is the solid in the first, often that lies under the tabloids. Ecause four minus X squared minus Weisberg While this parable Lloyd Z equals four minus X squared minus y squared, this intersect the X Y plane, which is the plane Z Po zero in a circle X squared plus y squared equals four or influential coordinates. This is R squared equals four, and then a Zara's positive. This implies articles, too, and so and so into a coordinates. Our region E If you set of triples our data Z, it's the fatal eyes. Well, because when the first often realize between zero empire but to are is going to lie between zero and two and Z will lie between zero The X Y plane and between the Paraiba Lloyd four minus X squared minus y squared. Reaching rectangular, rectangular or mixed. Switched to yeah, political coordinates is four minus R squared. And so the triple Integral um, experts y plus z over the region E. This is the iterated integral, which is integral from 80 to pi over two and drop Marco 02 in the Balkans equals zero for minus R squared of a function in terms of cylindrical coordinates. So this is our cosine theta plus our sign data plus Z 20 differential, Which for cylindrical coordinates, this is our times Easy BRD data within the anti derivative with respect to Z we get and they grow from zero to pi over to you go from 0 to 2 one back going out on our hold Arkan's ours are squared co sign data plus sign data eyes mm plus one half times are times Z squared from Z equals zero To see those four minus r squared DRD data with the anti derivative Sorry. Evaluating integral from zero to pi over two integral from zero to and this is or are squared minus art of the fourth Times Co sign the A plus sign data plus one half times are times or minus r squared, squared DRD theater and taking the anti derivative with respect to our this is integral from zero to pi Over two of here this is four thirds are cute minus 1/5 part of the fifth times Co. Sign data plus sign data and then this next term. Do the use institution in your head We get, see one half times negative one half times one third This is negative 1 12 times the inter function or minus r squared to the new power three from r equals 02 Data and evaluating you get integral from zero to pi over two and then plugging in. This is 64 15th times the cosine of data plus the sign of data and then plus and 12 times for cute, which is 16 thirds deep. Data taking anti derivative with respective data. This is 64 15th times, man man, today of your fear is going to be signed data minus cosign data. A 16 3rd data data equals zero to pi over two. Evaluating you get 64 over 15 times one minus zero plus 60 16 3rd times pi over two minus 64 15th times zero minus one minus 63rd time. Zero because zero and we simplifies to two times 64 15th 1 28 15th plus a thirds pi

Were given an integral and were asked to evaluate this integral by changing two cylindrical coordinates, So this is integral from negative to to integral from negative square toe four minus y squared to the square root of one minus y squared, integral from the square root of X squared plus y squared the tune UH, x z d z t X y personal of all notice that the region of integration is the region a bone the cone. The equation z equals the square e of x squared plus y squared or, in other words, Z equals our insulin tickle coordinates and is below the plane. Z equals two. We also have the wind lies between negative two and positive too, and X life between the negative square with four minus y squared the positive squared of four minus y squared. This describes in simple of radius too, centered at the origin in the X Y plane. Therefore, this in control integral from negative to deposit to integral from the negative square two for minus y squared Posit Square two for minus y squared. Integral from where we have X squared plus y squared to of our function xz dizzy D x Y making a switch to a wonderful coordinates. This is the integral from 0 to 2 pi integral from already closed zero two r equals Well, the code and the plane intersect when r is equal to two 02 and from Z equals. Now this is our too too of our function in terms of cylindrical coordinates. So this is our co sign data times E in the differential is now are B C D r d data. So regrouping begets integral from 0 to 2 pi integral from 0 to 2. Integral from our too, too. Um R squared co signed beta Z easy DRD data taking the anti derivative with respect cuisine. This is integral from 0 to 2 pi integral from 0 to 2 of r squared co sign data times one half z squared from Z equals R to Z equals two DRD. Data evaluating you get integral from 0 to 2 pi integral from zero to this becomes a lot of one half one half times coast are squaring coastline data times four minus R squared. He already data inconvenience theorem weaken right. This is a product of integral. So we have one half times integral from 0 to 2 pi data and integral from 0 to 2. Sorry, this is integral from 0 to 2 pi cosign data data and to go from zero to of four R squared minus arctic fourth er and taking into derivatives, we get one half times sign data from 0 to 2 pi times four thirds are cube minus 1/5 art of the fifth from 0 to 2 in evaluating. Obviously that our first factors signed data evaluated simply gives us zero so this whole integral is equal to zero.

This gay here gives us that zero lesson recall to ze less wrinkled, the nine minus r squared because x squared plus y squared is our square. And then these guys should be the other way. There should be d Y d X because otherwise this integral wouldn't make sense. Okay, And now this guy here is going to give us The zero is less than or equal to. Why which in cylindrical coordinates wise are science data is Lester equal to square root of nine minus X squared and cylindrical coordinates that squared of nine minus R squared co sign squared data and this whole thing if we square everything that we get our squared sine theta less or equal to nine minus r squared, close sign squared data If we add our squared co sign squared to both sides and keep in mind that sine squared Plus, Constance, what is one and we get our squared is less than or equal to nine. That gives us that zero is less or equal to our is less or equal to three. So now we need to figure out Seita. So I noticed that our X value was bigger than an equal to zero. We're sorry. This is Ah, we switch. This is D Y D X So sorry, R r Why value is going to be bigger than an equal to zero. So why value it's is positive. Okay, so that'LL tell us something about Fada. And then what about our X values While our exercise Khun b from minus three all the way up to positive three. So why values positive? But X value can be positive Or it could be negative. So the X values could be positive or negative. Why? Values and positive Then we should be thinking zero to pie, right? This is gonna be an appropriate choice of state. And now our X values allowed to get all the way down to minus are allowed to get all the way up to positive are And then the y value the biggest the Y value khun get is up to positive are small. Sticking. Get is zero case. Now we just have to tow plug this stuff in. So we have zero to pie. Zero three your tio nine minus R squared and then square root of X squared plus y squared. That's the square root of r squared says she's going to be our since ours Positive. And then keep in mind that when we switched to cylindrical coordinates, we always have to multiply by a another are So we're actually gonna have an r squared showing up here. And then this is easy D R. Data. Now it is Need toe, do some basic integration here and evaluate this. So yeah, keep in mind what you're integrating with. Respect to this is this first part's gonna be integrated with respect of Z. So we have nine r squared minus part of the fourth after we, you know, plug in this value in this value. Okay? And now we're integrating this thing with respect to our So I'll show a little bit more steps. Here we divide this guy. Buy three new exponents is two plus one, which is three. Here we divide by five new expo news four plus one, which is five. And this is evaluated from zero three data. Okay, so that's going to be some kind of ugly number. We'LL plug it and you should get something like one hundred sixty two united by five data and again, this is integrating with respect to theta, and this guy's obviously constant. With respect to theta, we get that which is just one hundred sixty two times pi divided by five.


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