5

EvaluateTII e? dV where E is enclosed by the paraboloid 2 = 4 + 2? + y? , the cylinder 22 + y2 = 4, and the xy plane_...

Question

EvaluateTII e? dV where E is enclosed by the paraboloid 2 = 4 + 2? + y? , the cylinder 22 + y2 = 4, and the xy plane_

Evaluate TII e? dV where E is enclosed by the paraboloid 2 = 4 + 2? + y? , the cylinder 22 + y2 = 4, and the xy plane_



Answers

Use cylindrical coordinates.
$$\begin{array}{l}{\text { Evaluate } \iint_{E} z d V, \text { where } E \text { is enclosed by the paraboloid }} \\ {z=x^{2}+y^{2} \text { and the plane } z=4}\end{array}$$

Z is bounded by X squared plus y squared, which is R squared. And four and four is going to be the upper bound here because that imposes bounds on R EF four is the upper bound and we get that ours between zero and two. If for was the lower bound, then our volume would be infinite. So we should know that our squared is going to be less than or equal to Z. It's going to be less than equal to four. And because of this, we get these bounds on our which is good, because we should be working with some finite shape here and then Goethe. We just want for dated and not repeat itself. So every angle just happen once so make theater between zero and two pi. Okay, so then the integral that we get is zero to two pi you too r squared to four Z and then multiply this by our times Easy, they are d theta and then from here is just doing some algebra wth So this should be equal The zero to two pi You too. One half disease squared R for Z has been evaluated from R squared up to four. So this gives us eight are minus one half part of the fifth D R data. And then we integrate this thing with respect Toe are we get zero to two pi eight, one half are squared minus one half one six hard to the six and this has been evaluated from zero all the way up to two. Then we get zero to two pi of this thing, which once we plug in to we get to squared, which is four divided by two, gives us two multiplied by a We had a sixteen when plugging a two over here we get minus sixteen over three, so this could, of course, simplified. But I'm sure everyone knows how to do that. And once we integrate this, then it's just whatever constantly had in here. This sixteen minus sixteen over three multiplied by to pie. And as you mentioned, this could, of course, be simplified. That's the answer

The problem is used. Cylindrical coordinates evaluated the triple integral. While he is a solid that lasts between the Sanders. This one on this one above XY plane and below The plane is equal to y plus four simply half this triple integral is equal to interior from 0 to 2. Pi interior from 1 to 4. Integral from zero to all right. Um, sign data. Us or exes are Hussein leader. Why is r sine theta times are you? See, you are the data, which is a hot you interior from 0 to 2 pi. And here from 12 or and our Squire co sign. Beta minus are square sign data palms R sine theta of or er the data, which is equal to the integral from 0 to 2. Pi 25 over or co sign for you to sign. Data minus 1/4. Has too far. Far. I'm sign data Squire. The data, which is a cultural 25 5/4 times negative force. Who assigned to theater from 0 to 2 pi minus one half arms through pie. US by force. Sign tooth data. Um, 0 to 2 high. And the answer is negative. Too far. Far pi over four

Okay, So this problem wants you to evaluate the triple integral off the square root of expert post Y square. Um, over the regional, be where he is. Inside the equation, the surface excerpts watched people 16 in between the planes, the Eagle data five and Z equals 24 All right, so the first step that we need to do is to convert everything into cylindrical coordinates. So the inter grown with the integral be well, you want to convert the tripling to go into some horn iss? Well, we know that our square equals two x squared plus y squared. And this integral is the square root of accurate plus y squared. So this is simply are And we also know that Devi is your government to our times. Easy d r. Di Fada. So this Yes. So this is essentially our square or square. We're taking the triple integral r squared. Now, um what else? Oh, yeah. We can also change the bounds boundaries here. Expertos y squared equals 16. We know that expert plus y squared equals r squared. So it's it's simply are able to four in the hornets and Z equals a zine cylindrical horn Essel. You can. You got the same next meeting to figure out the boundaries off this triple integral. So first we have easy. Well, easy is obviously just from there. 5 to 4. That's easy because you don't change from cylindrical. Come back Taylor to political. So the years from the Nevada For next we have er and we know that our has a maximum or four from our service extra plus y squared equal 16 or are able to four. So we know our radius r goes from zero to a maximum of four. And finally we have d theta, and this is just simply 0 to 2 pi because there are no restrictions that are going to cut the cut. Blake separate arts. All this off the service. All right, so now that we have are so now that we have our boundaries and our numbers and some coordinates next, when you have to just calculate it, Alright. So first we're integrating with respect. Is he going to grow our scare? With respect, Izzie is simply simply are square times e in the bounds are from four to negative five. Do you are be fatal and then the bounds for our and they stayed the same, just doing the first integral. All right, then, if we plug in the values off or a native, I would get, um, clinging with yet. R for R squared minus negative. Five R squared, which is nine r squared. So now we have to do a double integral off nine R squared DRG data. Where are is it is from 04 and the restaurants here are too high. Now we have to take the integral with respect to our So let's do that. Integral of nine are square with respective are last year's three are cube. So are well, it's going to get more space. Are integral is three are cube now and the boundaries is from 40 still on. We're doing it with respect to they don't nice. So if we were to plug in our boundaries, we would end up with the integral from You're not too high off. We aren't you are artists for and this is just 192 on respect to defeat up on Ben. Finally, if we integrated 109 2 with respect to with Spectra paid off on internal 0 to 2 pi. We would end up with an answer off 384 pie and this is your final answer.

The first thing that you have to do is figure out the bounds. So X squared plus y squared is sixteen, and our square is equal to X squared plus y squared. So this gives us that R is equal to squared of sixteen, which is four, and we're inside the cylinder. So we just have to be less than or equal to four, and our should be positive as well. And Seita not really any restrictions on theta, but we don't want to repeat ourselves. So this make they did between zero and two pi That way. No angles occur more than once, and the boundaries for Z are provided for us. Z is between four and twenty five, and then that's all their bounds. And then here we have square root of X squared plus y squared. So that's the squared of R squared. So that just gives us our So the integral becomes zero to two. Pi and Z Z, we said, was between four and twenty five, and then ah, we said was between zero and four. And then we have the square root of X squared plus y squared, which we said was R and then we have the You are FDR Easy Dee Fate which we have toe tack on to the end there. So this gives us integral from zero to two pi and a girl from four to twenty five. And this our times are gives us an r squared. So integrating that with respect are we get one third r cubed evaluated from zero up to four easy data. So that zero to two pi for twenty five, one third for cubed So four squared is sixteen sixteen times for is going to be sixty four So sixty four over three Easy the theta Okay, And now we're just integrating this constant with respect to Z. So this is just going to be that constant sixty four over three time Z Where's he has evaluated from four up to twenty five. Once we do that, we'LL have twenty five minus four here and then we have this d theta and again, this is just going to be constant with respect to theta. So once we integrate this, we have sixty for over three times twenty five minus four. So that's twenty one Time's data worth data goes from zero all the way up Tio to Pie, says his sixty four times twenty one over three times two pies. And then, of course, if you had a calculator than you could know multiplied these numbers together to get a simpler form than this. But this should be the correct answer.


Similar Solved Questions

5 answers
Civen UJuu Dulom entennc L IE oclertne nn unemt Enercd soluton PH 7.20O.RuOldicca #utch uecnbLorumyliVoluine of sock solution necdedVulutne 0f HCI nccded Adjust the Iml LuokVolum of HIC] nccrcd lo #iralcCTotal HICI ncededGubmul
Civen UJuu Dulom entennc L IE oclertne nn unemt Enercd soluton PH 7.20 O. RuOl dicca #utch uecnb Lorumyli Voluine of sock solution necded Vulutne 0f HCI nccded Adjust the Iml Luok Volum of HIC] nccrcd lo #iralc C Total HICI nceded Gubmul...
5 answers
Problem 4. Calculate the inverse A-1 of the matrix ~2 ~2by Gauss elimination; using the formula for the inverse.
Problem 4. Calculate the inverse A-1 of the matrix ~2 ~2 by Gauss elimination; using the formula for the inverse....
5 answers
Find sets parametric equations and svmmetric equations the line through the point parallel Point Parallei (-3,7,2) X-= 2ar-5the given vector(if possible).(a} Parametric equationsSymmetric equations 4 42.4". 42.Y3-1-2 41.27422+2
Find sets parametric equations and svmmetric equations the line through the point parallel Point Parallei (-3,7,2) X-= 2ar-5 the given vector (if possible). (a} Parametric equations Symmetric equations 4 42.4". 42.Y3-1-2 41.27422+2...
5 answers
Calculate the pH of HF solution having concentration of [H3O*] 1
Calculate the pH of HF solution having concentration of [H3O*] 1...
5 answers
Deep 'erential ec u st Sxs,6(4+v) the3Finc ( eSu&rSepar 0b le Bernaul/iexact homogenaus neevr
deep 'erential ec u st Sxs,6(4+v) the 3 Finc ( eSu&r Separ 0b le Bernaul/i exact homogenaus neevr...
5 answers
Given below is the figure of a sarcomere. Identify the labelled as $A$ to $D$ and select the correct option. B I
Given below is the figure of a sarcomere. Identify the labelled as $A$ to $D$ and select the correct option. B I...
5 answers
For which nonnegative integers $n$ is $n^{2} leq n ! ?$ Prove your answer.
For which nonnegative integers $n$ is $n^{2} leq n ! ?$ Prove your answer....
5 answers
6x3+X^_yZ flx, y) = if (x, y)= (0, 0) 3x2+y2 if (x, v) - (0, 0)tien fx(o, 0) =0 2D 7213does not exist
6x3+X^_yZ flx, y) = if (x, y)= (0, 0) 3x2+y2 if (x, v) - (0, 0) tien fx(o, 0) = 0 2 D 72 13 does not exist...
5 answers
1222 dx 36Evaluate the integral:(A) Which trig substitution is correct for this integral? 36 sec(0) 6 sec(0) 6 sin(0) 36 sin(0) 6 tan(0) 36 tan(0)(B) Which integral do you obtain after substituting for € and simplifying? Note: to enter 0, type the word theta_ de(C) What is the value of the above integral in terms of 02(D) What is the value of the original integral in terms of 1?
1222 dx 36 Evaluate the integral: (A) Which trig substitution is correct for this integral? 36 sec(0) 6 sec(0) 6 sin(0) 36 sin(0) 6 tan(0) 36 tan(0) (B) Which integral do you obtain after substituting for € and simplifying? Note: to enter 0, type the word theta_ de (C) What is the value of the...
5 answers
Quetlion 47wich comntynaliun rosuls In Ine most ulieclive Atei Kcuncle0 0.25 KNOjand 0 20 HNO)NH Cland E0 M NH}060 MK SO; snd 0.70 MKNSO40 0 00j M HC;HsOz (Denzor; jcid) ana 0.40/ Nac-Ho
Quetlion 47 wich comntynaliun rosuls In Ine most ulieclive Atei Kcuncle 0 0.25 KNOjand 0 20 HNO) NH Cland E0 M NH} 060 MK SO; snd 0.70 MKNSO4 0 0 00j M HC;HsOz (Denzor; jcid) ana 0.40/ Nac-Ho...
5 answers
Faricee Valerle Shah ale vislllng Hawall; Ile ! awallan Cullural Center Honolulu; they are told thal 2 oulotu Davld Barnes and group 21 people Tundomly picked fl free lesson ol a Tahilian dance probability Ihat bolh Davild anc Valetle plcked for Ihe Tullllan dance lesson? (Round your answer t0 declmal What Me pies0d |LeqaloltyWlral Is the probability thal Valerie gets plcked before Davld for the Tahifian dance lesson? (Round your answer t0 declmal Jaces |
faricee Valerle Shah ale vislllng Hawall; Ile ! awallan Cullural Center Honolulu; they are told thal 2 oulotu Davld Barnes and group 21 people Tundomly picked fl free lesson ol a Tahilian dance probability Ihat bolh Davild anc Valetle plcked for Ihe Tullllan dance lesson? (Round your answer t0 declm...
5 answers
A) An electric potential of 20.0 Vis applied acrOSs gauge , 1.0 m long copper wire. What effect does this have On the charges inside? b) Determine the resistance of this Wire and the current Neglect any changes in temperature: Determine the drift velocity of the electrons inside of the wire d) The entire length of the copper wire is coated with 1.0 mm thick plastic which has resistivity of 0.9x[0 Qm Calculate the radial resistance of this plastic.
a) An electric potential of 20.0 Vis applied acrOSs gauge , 1.0 m long copper wire. What effect does this have On the charges inside? b) Determine the resistance of this Wire and the current Neglect any changes in temperature: Determine the drift velocity of the electrons inside of the wire d) The e...
5 answers
If the Cartesian coordinates of a point are given by (-2,Y) and its polar coordinates (5,1208). The r and y coordinates of the point areSelect one: a.r=-8, y-7.28 b.rz4,y-3.46c.rz4,y=-1.95 d.r=-6,y-3.46
If the Cartesian coordinates of a point are given by (-2,Y) and its polar coordinates (5,1208). The r and y coordinates of the point are Select one: a.r=-8, y-7.28 b.rz4,y-3.46 c.rz4,y=-1.95 d.r=-6,y-3.46...

-- 0.024343--