evaluate-the-following-integrals-in-cylindrical-coondinates-the-figures-if-given-illustrate-the-re-7

Question

Evaluate the following integrals in cylindrical coondinates. The figures, if given, illustrate the region of integration.
$$\int_{0}^{3} \int_{0}^{\sqrt{9-x^{2}}} \int_{0}^{\sqrt{x^{2}+y^{2}}}\left(x^{2}+y^{2}\right)^{-1 / 2} d z d y d x$$

Issa Dababneh · Accepted Answer

so and this question were asked to be very rate this triple, which is a triple integral of X squared plus y squared to the power of negative half. Where&#x27;s he goes from? Zero to the square root of X squared plus y squared. Why goes from zero to the square root of nine minus X squared. Expose from zero. Okay, now what do we first do? So since we&#x27;re going to cylindrical coordinates, we know that expert. That&#x27;s why square it is important for square. So this right over here in this bar square. So we have our squared to the power of negative one hand. But also here we have expert plus y squared, which is also R squared and the square root of our swear it is just part. So our C goes from zero to our Now we know that our why goes from zero to the square root of nine minus exploring our X goes from zero. So I tried toe brought this. So this is my ex. This is a lie. So first we have that expose from 02 threes of I plug in zero for the square root of nine minus x squared. That means that zero I get square root of nine, which is simply three. And now, when excess three law is going to be the square root of nine minus three squared or zero. So basically, I have this border over certain. So we see that our why goes sorry are our boats prophecy. Oregon three Vilar Beta goes from zero to Firebirds. So next time you have the are we can see that our our goes from 0 to 3 by large FINA goes from 0 to 5/2. All right, now, just one extra thing. This volume element to go from rectangular two cylindrical port. It&#x27;s it&#x27;s are easy artist. So we have to add this extra are okay now are square to the power of negative 1/2 twos. Cancels are left with our to the negative one Now are to the negative one times ours one. So, basically, are you triple integral? Is this right over here? So now what&#x27;s the integral of, uh, easy? Hold us to see where&#x27;s the ranges from? Zero are. So if we do, our minus your we just get on. So now we have to do this double integral over here. So the double integral of our where argo, where are goes from 0 to 3 and data goes from 0 to 5. Now, the integral of are the oranges are squared, divided by two are limited integrations from Syria. So if we plug those and we get nine square, give a three swear divided by to buy This year of squared divided by two which is just nine so nine have the state nine has is a constant. So now we think the integral with respect data&#x27;s just gonna be nine house time state up where if they don&#x27;t goes from zero to pile with too. So they plug that into get 9/2 times higher for to my zero, which is simply nine pie divided by four. So the value of our triple integral is nine pie divided by four

Issa Dababneh · Answer

so this question were as deserving the fall in Tripoli. So the first thing we&#x27;re gonna do is change this to cylindrical court. It&#x27;s so that means that X squared plus y squared here we&#x27;re can change it to r squared throughout the square root of r squared. Okay, now, RZ so are easy right over here. Course goes from negative one tool. So we&#x27;re gonna put our easy here first, and I&#x27;m gonna have it also. Now, the trick is determining what are our or in stato. So next, see, if we draw Why is equal to X is equal to the square root of Yeah, three wine and we draw X is equal to the square root of one minus y squared. What we get is this Rydell. So we see, like a semi circle intersecting with a lot. So at zero. So our radius. So we see that this is a circle off radius one. All right, great. So are our goes from zero No one. Now, what about our data? Well, so we know that, uh, why goes from 0 to 1/2? Because right over here we have that Why goes from 0 to 1 and then it zero, we have 00 at 1/2. We&#x27;re gonna have um yes. So this is 1/2 right over here. And then at 1/2 we&#x27;re gonna have what X is 1/2. Why is the square root of three divided by to all right? Now, what is this angle? One angle gives us a Y value off square root of 3/2. Well, if we think about it, we know that sign. 60 square root of three divided by. So we know that our angle goes from 60 60 degrees or high over three, and then it ends. So goes from here all the way to here. So it goes for high over 32 pi over. So now we arranged everything. Correct. So we have ours. We going from negative 11 Our are going from 0 to 1 on our fate of going for high over three empire with two. But there is one extra bit that we need to remember. So remember to go from your volume element in the rectangular coordinates, which is easy the x Bortz to your volume element in sunder Coke ordinance. You have to tack on you have to add. Or so it&#x27;s our Deasy. The organ. All right, Great. So now we have Our times are is are square. And now we&#x27;re gonna take the integral of r squared with respect. Izzie. But then R squared is a constant. Just so we have our square time. See, where&#x27;s he goes for negative 1 to 1. Now, if we take the difference, one minus negative one is to. But now we have the double integral, uh, of two r squared. Uh, where are goes from? 0 to 1. And data goes from five. Over. Frito Pie. Over. Okay, great. Now will you continue? So if we take the integral to our squared with respect to our debts to our cube divided by three, where are goes from 0 to 1. We take the differences just to third vital zero, which is 2/3. Now. We have to take the integral to third with respective data for a state. Arranges for Firebird. Frito pie over to. That&#x27;s just two. Divided by two by three times data Where faded rain just for fire for Frito Pie. Over to now we take the difference off by over two minus pi over three. We&#x27;re gonna first create a common denominator. And then we get to over three times higher for six. Well, that means that our triple integral is equal to pi, divided by

Issa Dababneh · Answer

All right. So this question were asked to determine the triple integral off E to the minus X squared minus y square. Where why ranges from X to the square root of one minus X squared x ranges from zero spurt of two divided by two and CIA ranges from zero. So basically this triple integral right over. So the first thing we&#x27;re gonna do it&#x27;s right business. According it. So it&#x27;s gonna be he didn&#x27;t power all. If we pull out a negative, we have negative X squared, plus vice versa. That&#x27;s just negative force. All right, now, let&#x27;s see. Where are Ziering? So are see, Like we said above it goes from 0 to 4. So, uh, values were zero from zero. All right, now let&#x27;s see where what? Where are our bring? Just from what toe? Well, we look at the first term you want There. We have right over here. We see that it goes for X to this word off. One minus expert, and then we see the integral before that, or the D X, which is right over here that are X ranges from zero to square root of two. Well, that should make sense to us. So when x zero Oh, why is gonna be one minus a square? Root off one, Just just one. All right. So obviously these values. So if you go from zero discriminative two divided by two and why is once word of one minus expert, we immediately know that this is just a unit circle, right? So our our ranges from 0 to 1 has this right here. We realize it&#x27;s just a unit, sir. All right. And we also know that our X ranges from zero were routed to a divided right. So I know that X is our co sign. So for what value off X Do get square root of two divided. Well, we get where root of two divided by two it Hi over four. Correct. So we realize that our d state of simply ranges from zero to pile. All right, so we transformed everything into. Now the last thing we have to remember is the volume element we have to put at bar. So it&#x27;s r DZ Ortiz. Either you have to tack on for So now we set everything up when we&#x27;re ready to go. So now we integrate so how to read. Okay, so that&#x27;s first of all, we&#x27;re gonna take the integral of our E to the negative are square with respect is easy. But what do we know? This does not contain any disease. So since it does not contain any disease since it does not contain any disease, the integral visitors are e to the negative r squared time seaworthy ranges from zero now four minus zero is just four. So now we have to take the integral from 01 of four times R e to the negative r squared your so this integral but I but a green box So this interval is gonna take us some time, but we&#x27;re gonna do it right now, so I colored it in green. So let&#x27;s try to integrate no free net. You quote negative r squared. We noticed that you is negative to art. That means are the ours does you divided by negative. So now we plug everything in. So we get four divided by negative to eat of the U. T. And the last thing we have to do is that we should change the limits of Integrys. So if we take our zero we plug it in for our we get negative zero square zero that if we take one, we get negative one squared. Or just so these are limits of integration, but since we already have ah, negative. Right year weeks weakened. Switch it. So the limits become from negative 120 of to you. You are the integral to you if you because the exponential function has property where the integral of the derivative of he&#x27;s just okay. And now our limits of integration are from negative 1 to 0. So we plug those answer, get to either the power of euro minus two me to the negative one. Well, eat it is yours, Miss one. So we have two minus two divided by Eve. We pull out too, as a common factor. This is what a working girl so are integral off peak inside. It&#x27;s simply too Times one minus to the power. Okay, so now all that we did this re happy, girl, that cup. And now we have to take the integral where we and yeah, so we go back up. So now we plug this in. Now, this is a constant. So he&#x27;s just the number 2.71 whatever. So now it protect the integral of this. With respect to data, we just get two times one minus one, divided by E time state aware Fada ranges from zero to fire. Before that would look to that end, we get we get high over four minus zero, which is just by over four. Now, finally, we can simplify so two pi divided by forces pi over two. So the value of our triple integral is just high over two times one minus one, divided by

Issa Dababneh · Answer

So in this video were asked to determine the following to grow. And they&#x27;re looking at this interview over at this triple integral really realize it&#x27;s a whole bigness. And the reason for that is that this is, uh, rectangular coordinates. So basically, what we have to do is we have this graphically were presented like this. We&#x27;re just trying to find the area 1/2 a coat. Okay, so now let&#x27;s look at. So we try to right the integral That&#x27;s underlined in blue in polar coordinates. Sorry. Cylindrical coordinates we have that are is equal to the square root of X squared plus y squared. So we have that Arthuis ranges from 0 to 9 minus three. Or and then we also have that our data goes from zero to pi. Right? Because our our beta goes from zero all the way to pie, and then our radius. Of course, we start. We have our radius goes from 0 to 3. So this is our radius right over here. All right, so now we know that the volume element in cylindrical coordinates instead of DZ dx dy lie, it&#x27;s our thes e d data. All right, now that&#x27;s begin. So the integral off our with respect busy, there&#x27;s just or times because our is just the constant with respect hours the ranges from 0 to 9 minus three or so. If we put that in, we have nine minus three R minus zero, which is there simply nine minus. Really? All right, so now we need to take the integral off Far times nine minus three. Or do it. We can expand, we can distribute. This are becomes nine ar minus three are square time. Now we&#x27;re gonna find the integral well nine ar minus three. Our spirit is just a constant with prospective data so now are integral becomes the integral from 0 to 3 of nine ar minus three R squared times data where fada ranges from zero. So pi minus zero is just high. So finally, our final inter groans from 0 to 3 of high times nine are mines. Three r squared D are now if we take this into grow becomes high times nine r squared, divided by two minus art and our our ranges from 0 to 3. Now, if we plug these in zero and three, we get high times nine times three squared, divided by two minus three cubed minus zero. So we can just forget about this. So nine times nine is 81. So apply times 81 divided by two, minus 27. And if you do the math, you get 27 divided by two times plight. Okay, so now we can, uh, check this. So remember this. We&#x27;re just actually finding the volume of a cylinder that has Radius three and fight night. So we know that the volume. So now we&#x27;re just checking our answer. It&#x27;s 1/3 high R squared each, which is just 1/3 times high. Our Oreste three. So this is three squared times. Each is nine. So now what do we get we get now? This three of this cancels. So we get 27. Hi. All right, Now, this is the volume for a whole coat. Now we want half of that, so the volume is gonna be 1/2 times 27 pi because we have half a cool. So this is just 27 divided by two by so both ways, the weekend check our answer. And then the the triple integral is basically 27 divided by two heart

Issa Dababneh · Answer

So in this video, we&#x27;re asked to determine this triple crow right over here. And we&#x27;re told to use, uh, still in jerko courts. So the first thing we do is we notice that this x grand plus wise, where there is simply are spurred so we can substitute that. And we also know that the volume volume in rectangular coordinates to change it into cylindrical coordinates is our easy, uh, our yard. But now we have to arrange our easy used yours and staters in the correct order. Cern, etc. First, the first thing we notice better see goes from is your room to to So our C goes from 0 to 2 and are easy comes first. Okay, Now we&#x27;re are goes from our radius goes from 0 to 3. And are we arranged? So from 0 to 3 and then we put dealer after And finally, our data, our angle goes from zero overly seven years. So how we have zero pie and the d stayed up comes in. All right, so now we arranged everything in the correct order on. Now we proceed. So we see that, um so if we take the integral are divided by one plus our skirts are squared. The prospective Z that&#x27;s just are divided by one plus R square times where Z goes from 0 to 2. All right, great. So now we subtract so to a minus zero is just too. So that means now we have to evaluate the integral the double. In Europe, I have 03 two are divided by one plus ours where the RDC. So now the box and green will evaluate it right next to it. So this is the box in three. All right, so we let u equal one plus r squared you is to argue. So now we can rewrite are integral, as do you divided by you and the limits of integration becomes a is from 1 to 10 wife. We plug in Europe for our we get one plus zero squared, which is one I&#x27;m a rip lug in 34 are we get one plus nine. So this is our, um to grow and we know the integral BU divided by use. Just a natural law of you where you ranges from 1 to 10. Now, if we plug that in, we have the natural log of 10 minus a natural law one. But the natural longer one is so that means are integral is just the natural log. 10. All right. Great. And remember, the natural laws of 10 is just a constant. All right, so now this is the integral to evaluate our final integral. So the integral the natural log of 10 data is just data. You know, we have the natural log of 10 times data where data ranges from zero to pi. We take the difference. Hi. Minus. You&#x27;re just fine. So that means our triple integral is high times the natural law.

Issa Dababneh · Answer

So in this video, we&#x27;re asked to be triple integral. Uh, what we have here still along to say and, uh, what we can do. First we notice an X squared and the square and then also the inmate. The grab that were given is that of a stolen. So So we&#x27;re gonna use cylindrical coordinates, so we know that. So if we look at our image at our picture, see that, um, the Z goes from a negative 1 to 1. So along this, since we go from negative 1 to 1, that&#x27;s our first limit. And then we also noticed that the radius goes from 01 because of surrender has a radius of 10 Tawana that would put RTR and our data. We&#x27;re going in circles. So say that goes from 0 to 2 pi. All right, great. So now we arrange their easy VR correct blend. Remember the volume for we need to tack on a farm. So we have our easy New York data in regards to the order. All right. We also have in the inta grand X squared plus y squared, but we know in cylindrical coordinates X squared. Plus y I swear to simply are square So we have our square to the power of rehabs times for well, this in this cancel. So we have our cube times are so this is our integral are cute times are it&#x27;s just harder the power for and now we have to take the integral part of the power or with respect is well with respect to C c r To the power for is a constant So now we have this. But this doubling to grow for the integral DC is just so really our to the power of four times you were C goes for negative 1 to 1 now one minus negative one is to So that means our double integral is the following. So the integral from 0 to 2 pi at the integral from 012 r to the power for the or now what&#x27;s the integral of two times are getting power for with respect to all? Well, that&#x27;s just to r to the power of five divided by five, where are varies from one. Now if we plug in zero and what we&#x27;re gonna get to to the power 5.0 are just two to the power. Now we take the integral of this with perspective data. Since it&#x27;s a constant, just do their power. Five times data where data varies from 0 to 2. Pi now 3 to 5 minus zero. We get two pies on to two divided by five times to pipe. Well, that&#x27;s just for pie. Divided by funny. So the triple integral is four pi divided by

Problem

Evaluate the following integrals in cylindrical c…

Problem 21 Easy Difficulty

Evaluate the following integrals in cylindrical coondinates. The figures, if given, illustrate the region of integration.
$$\int_{0}^{3} \int_{0}^{\sqrt{9-x^{2}}} \int_{0}^{\sqrt{x^{2}+y^{2}}}\left(x^{2}+y^{2}\right)^{-1 / 2} d z d y d x$$

Answer

$9\pi/4$

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William Briggs, Lyle Cochran, Bernard Gillet

Calculus Early Transcendentals

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$9\pi/4$

Calculus Early Transcendentals

Anna Marie V.

Heather Z.

Kayleah T.

Caleb E.

Vectors Intro

Vector Basics - Example 1

William Briggs, Lyle Cochran, Bernard GilletCalculus Early Transcendentals

Anna Marie V.

Heather Z.

Kayleah T.

Caleb E.

William Briggs, Lyle Cochran, Bernard Gillet

Calculus Early Transcendentals