5

2. (Spts) Evaluate Ilf zdV is the solid that lles above the XY-plane; within the cylinder x2 + y2 = [,and below z = x +y_...

Question

2. (Spts) Evaluate Ilf zdV is the solid that lles above the XY-plane; within the cylinder x2 + y2 = [,and below z = x +y_

2. (Spts) Evaluate Ilf zdV is the solid that lles above the XY-plane; within the cylinder x2 + y2 = [,and below z = x +y_



Answers

$17-28$ Use cylindrical coordinates.
\begin{equation}
\begin{array}{l}{\text { Evaluate } \iiint_{E}(x+y+z) d V, \text { where } E \text { is the solid in the }} \\ {\text { first octant that lies under the paraboloid } z=4-x^{2}-y^{2} \text { . }}\end{array}
\end{equation}

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

All right. So we have a little more complicated problem here, so we wanna have the salad outside. It's outside. Uh, X squared plus y squared is equal to one. Um, and then bounded above by Z z equals negative X squared plus y squared plus eight and below by Z is equal to x squared plus three y squared. So we're gonna have a little, um, complication where you usually do. It's not really that hard to deal with. We'll just have to worry about this one part here. So if we go ahead and start drawing our our 30 space, um, we'll have Here we have this problem, Lloyd, that's gonna be facing down. And then over here gonna have is probably facing up, and they're gonna intersect. At a certain point, that doesn't really matter where, however, usually, if we would include this area this entire volume, it only include two volume outside of this still in there. So we're gonna have a sort of cylinder cylindrical hole chopped out the center, so it's easy to deal with. We'll just have to find whether region or is our region that we're integrating over and just not include this area This this part of the region. So we know that z one Soviet Z one z two. We know that Dizzy once created this YouTube for our points. Now we just want to set him equal to each other to find out what uh, it could be what are regional would be. So we want to said Z one is you got to see to and we end up with a negative X squared plus y squared plus eight is equal to expert plus three y squared. We do some algebra we can get a is equal to x squared are two x squared plus two y squared, leaving us with a familiar circle equation X squared plus y squared is equal before this means that her intersection is that r is equal to two, which is a circle of radius to and polar hornets. And you want to make sure that our are far area integrating over is outside of our secret of one. So, um, you get a pretty good big clue of over doing here. So we know that for our region that were integrating this over, we have radius has to be less than or equal to two since we're starting off with our outside boundaries, um, R is equal to and our inside boundaries are Zagato once. So instead of going all the way to zero, we just end up at one. And, as usual, BB integrating over 0 to 2 pi from data. Now we go ahead and start doing this. We just take a double winning roll over our of C one minus he to Okay. And in this case, it's the one of over our of Ah, we have eight minus two x squared, minus two y squared d A. And in polar coordinates is his 0 to 2 pi she owed to actually want to. And, um, girl from of eight minus two R squared. Already already. Data now separating our generals Benevolent nated intervals. You, too, Polly of data. And in a girl from 1 to 2 of eight R minus two R cubed we are, as usual, went up with two pi out front times. Four r squared minus 1/2 are to the fourth from 2 to 1. And now it is up to vary with this, we end up with ah, four times two squared or 16 minus 1/2 times two squared, which is 1/2 times 16 or eight and then we'll have four minus 1/2 that's minus four minus one have, which is 7/2. So our final expression this two pi times eight minus 7/2 or pi times 16 minus seven. Living us with nine pi as our final line.

All right. So we want to find, um the volume of the salad that's outside of the cell in their ex cripples white skirt is one, uh, with the top bound being the sphere X scripless y strip of the square is eight and the bottom being thus, um, thats cone grow. Aziz, you gotta square root X scripless y squared. So first thing you want to do is convert all this polar. This is ours. You go. One, this is Z square is equal to a minus. R squared. And this is disease equal to our here. You can correct us further to Z is equal to square root hate minus r squared. So now we have our top Z are bottom Z and our region on the outside. So we know the the woods outside and we'll find the other border. So we want to find the intersection of these two ah services. So we have the surfaces. Um, we have running Malan Z we have Do you want is equal to see two or square root of eight. Minus R squared is equal to our going ahead and, um, taking square beside to get eight minus R squared is equal to R squared or, um, bringing are over here. Dividing my to get r squared equals four or r squared R r is equal to So we know that our outer bound is ours, able to our inter bound is ours a little one. So we know that our our surface is actually in between two circles one article one and article two. And since their circles, we know that data must go from 0 to 2 pi. So to find the volume of this, we need to take the top surface mines the bottom surface and pretty clear that, um, z one is granted and see too. So we want to find double integral over our of Z one minus he to yea or in this case, and grow from 0 to 2. Pi and girlfriends here are from 12 of square root of eight minus R squared. Plus, uh, sorry. Minus are are the rt data. Go ahead and bring this out. Okay, everyone, 0 to 2 pi data integral from 1 to 2 of our square root of eight in minus R squared. Minus R squared. We are, um, this first integral to do a U substitution of eight minus r squared. We get the u is equal to negative. Two are the are So we end up with the integral. Um, we get minus one integral of square root of you, do you or in this case, we get 2/3 times one hat So negative 1/2 times 2/3 you tooth rehabs or native 1/3 you to three halves. So we get the first interval gets we get this thing is two pi then we get this is native 1/3 times a minus R squared to the three halves minus two are or sorry are cute over three from 1 to 2. Um, we can go ahead in fact, at a negative. Want there to get its native to pirate three times integral from 1 to 2 of eight minutes or so are not interval but evaluating this quantity from one into. And if he violated that too, we get negative to power three times. Um, eight minus two squared is four. So forth to three halves is to Cuba. Great plus r cubed is in other aids. Then we get, um, eight minutes r squared over seven 23 halves on minus one. So overall, we get on the following. We get negative. Two pi over three times. Um, 16 minus five is a 16 minus one is 15 minus seven. Discover to seven. Or if we had, uh, but the negative Through we get two pi over three times seven. Screw to seven, minus 15. And just to make sure this is ah valid and greater than zero, we can go ahead and find this, and it equals approximately, um seven point for so we know that, um, this is a plot. Ah, Valid answer to our problem.

In the question were to use the level integration to find the volume of this fuller the solid in the first up in Monday, above the parable life off their nickels. Two X squared plus three y squared below by the plane ready, close to the road and literally my wyffels access. Where? Hand. Why calls? Thanks Now moving towards dissolution Yeah, William below the function that equals two ex lie and above the region is given by being noble integration with the original. If it's why do you so the volume can americanus double invigoration over the region are X Square plus revised where the which and American as integration from zero Kalanick configuration problem. It's Rex exist were less rely Square de y. There's no First, we will be solving engineering, keeping their learning regulars that yes, we were excess. Why last? Why you definitely by employees who exist like where? Rex B. X. In these limits, we will get integration from 01 Let's do minus 6 ft minus six with 56 b. It's no. So when this integral we will get next to depart Hora phone who minus X to depart five fight my next extra seven upon standing in regulation going from zero colon and you will get 11 by 70. This will be the final answer for the question.


Similar Solved Questions

4 answers
,9(x) = ~x+2,hkr) = x_ 7x+ 10,andj(r) = 4r3 + 3r2 _ 2x +1. N.Letflx) = x2 +5 ,_ solutions; points each) Find the following: (Show complete16. (f + 9)(x)17. (j- h)()(f* 9) (x) 19. (j+f)(x)20. ()c)21. (he g)(+) (g e f)(r)23. (f e 9)()24. (he 0(2)25 (j " 9) (3)
,9(x) = ~x+2,hkr) = x_ 7x+ 10,andj(r) = 4r3 + 3r2 _ 2x +1. N.Letflx) = x2 +5 ,_ solutions; points each) Find the following: (Show complete 16. (f + 9)(x) 17. (j- h)() (f* 9) (x) 19. (j+f)(x) 20. ()c) 21. (he g)(+) (g e f)(r) 23. (f e 9)() 24. (he 0(2) 25 (j " 9) (3)...
5 answers
The sequence fn is given as fi t2020te =3, fn+2 = fn + fn+l forn > 1. How many even terms are between the first terms of that sequence?673B674101010111347
The sequence fn is given as fi t2020te =3, fn+2 = fn + fn+l forn > 1. How many even terms are between the first terms of that sequence? 673 B 674 1010 1011 1347...
5 answers
NHzNHzNHzACIsSynthesis?OhOzN
NHz NHz NHz ACIs Synthesis? Oh OzN...
5 answers
Consider the function whose graph is the surface shown at the right:Which traces g0 with which graphs below?1. The trace for y = 01.00 0.00 L.002. The trace for x = 0.23. The trace for y = 15The trace for x = 1.8Drag the surface to rotate
Consider the function whose graph is the surface shown at the right: Which traces g0 with which graphs below? 1. The trace for y = 0 1.00 0.00 L.00 2. The trace for x = 0.2 3. The trace for y = 15 The trace for x = 1.8 Drag the surface to rotate...
5 answers
Given: (q is number of items) Demand function: d(q) 216.6 0.2q Supply function: s(q) 0.4q?Find the equilibrium quantity:ItemsFind the equilibrium price:Next Question
Given: (q is number of items) Demand function: d(q) 216.6 0.2q Supply function: s(q) 0.4q? Find the equilibrium quantity: Items Find the equilibrium price: Next Question...
5 answers
Use a graphing utility to graph the function and to approximate any relative minimum or relative maximum values of the function.$$f(x)=x^{2}-6 x$$
Use a graphing utility to graph the function and to approximate any relative minimum or relative maximum values of the function. $$f(x)=x^{2}-6 x$$...
5 answers
Points) Evaluate the double integralJI p(s,y) dApoints) Interpret the result in Part(b).
points) Evaluate the double integral JI p(s,y) dA points) Interpret the result in Part(b)....
5 answers
4.6, Derive expressions for the group velocity acceleration (dvg/dt ) . and the elfective mass (m of electrons using the E-k relation for the tWO- dimensional square lattice described in Problem 45. If cos(k,a/2) = plot E, "g. dvg/dt , and m versus k for the One-dimensional ([-D) crystal altice_
4.6, Derive expressions for the group velocity acceleration (dvg/dt ) . and the elfective mass (m of electrons using the E-k relation for the tWO- dimensional square lattice described in Problem 45. If cos(k,a/2) = plot E, "g. dvg/dt , and m versus k for the One-dimensional ([-D) crystal altice...
5 answers
QUESTION 13It is known that among 30,000 absentee ballots, candidate X has 20,000 votes: In a test of quality control, 100 of the ballots are sam- pled at random. Suppose the binomial distribution applies to the number going to candidate X and using the normal approximation to the binomial, find the probability that 70 Or more of the sampled ballots are for candidate X_
QUESTION 13 It is known that among 30,000 absentee ballots, candidate X has 20,000 votes: In a test of quality control, 100 of the ballots are sam- pled at random. Suppose the binomial distribution applies to the number going to candidate X and using the normal approximation to the binomial, find th...
5 answers
7 -322 32
7 - 3 2 2 3 2...
5 answers
Q11 5 PointsWhat is the molarity of 0.18 L CaCI2 solution that contains 3.1 moles of CaCI2? 0 31M 0 0.56 M 0 17M 0 3.3m
Q11 5 Points What is the molarity of 0.18 L CaCI2 solution that contains 3.1 moles of CaCI2? 0 31M 0 0.56 M 0 17M 0 3.3m...
5 answers
Hhnle Eltsik ol tr Lrp E m Posian MUXE4 [4u-I47. 02 oudu tn- "pdieprn' (al 4dk TotadIR hxkam u[hrr FetIknar Rrch Untfuad bazur tk E{ menedi [email protected]&kp / Kro Justily )Lnt Mnttt,Ethean gnnn (or mwnil dhnbrnl lotu p =utLl len t LATI mtang _ cunyml traed uten Ikr koceumilan Emriu4 IatEe In EIt 41LC3 L. Ie = FHEnkall ArTrLIOAh !Fn IE' Lnumurniluk4c Luteu4 Lur nLarucE [uitc_vite Iru W Laltxudr Iqur 4447 4lr LeT Ma"Aa
Hhnle Eltsik ol tr Lrp E m Posian MUXE4 [4u-I47. 02 oudu tn- "pdieprn' (al 4dk TotadIR hxkam u[hrr FetIknar Rrch Untfuad bazur tk E{ menedi [email protected]&kp / Kro Justily )Lnt Mnttt, Ethean gnnn (or mwnil dhnbrnl lotu p =utLl len t LATI mtang _ cunyml traed uten Ikr koceumilan Emriu4 IatEe In EIt...

-- 0.018865--