5

Use cylindrical coordinatesEvaluate the integral, where E is enclosed by the paraboloid z = 6 + x2 + y2, the cylinder x2 + y2 = 9, and the xy-plane. ff dV (e14_ 8 -...

Question

Use cylindrical coordinatesEvaluate the integral, where E is enclosed by the paraboloid z = 6 + x2 + y2, the cylinder x2 + y2 = 9, and the xy-plane. ff dV (e14_ 8 - X

Use cylindrical coordinates Evaluate the integral, where E is enclosed by the paraboloid z = 6 + x2 + y2, the cylinder x2 + y2 = 9, and the xy-plane. ff dV (e14_ 8 - X



Answers

Use cylindrical coordinates.

Evaluate $ \iiint_E (x + y + z)\ dV $, where $ E $ is the solid in the first octant that lies under the paraboloid $ z = 4 - x^2 - y^2 $.

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 of X over the region. What used bounded by the horizontal plane Z equals zero and the slanted plane Z equals X plus y plus five. And it is bounded by two cylinder services. One expert with y squared equals four, and the other extra plus y squared equals two of six. All right, so the first thing we need to do, as always, is to convert everything into cylindrical coordinates. So let's do that. We're gonna have our three into girls, and first we have X. So what is accidents on your corn is? Well, that's simply just our times coastline of data. Right? And Devi and senator cornices are times DZ de are de fate also, you know, a square to er and right easy G R D theta. Okay, so next let's look at the boundaries. So the boundaries, the first one is equal. Zero has a horizontal plane. If you convert that to some of the hornets, well, that's just Z equals zero. So not much has to be done there. However, the next plane Z equals X plus y plus five can be converted to salute jewel coordinates by replacing X and y by with are cool certain data and are assigned data. There you go. And then and then the bottom tomb boundaries are cylinders, and we know that extra plus y squared equals two R Square. So we can simply rewrite these, as are able to and r equals a three. All right, so now we have everything and converted into so little hornets. Now the next thing we have to do is find the boundaries. So first we have here is DZ so easy we know that the bottom the bone boundary for Z is the equal zero because I was the lower horizontal plane and the upper plane is the equals two. Arcosanti oppose our Senate. Taylor was fine. So we can write that as our upper boundary for Z or Co. Santana plus plus honor science data plus fine. Next we have the R and looking at our two cylinders, we have our egos 2 to 2 articles to three so we can write those as our boundaries on the interval for D. R and finally decided on Well, that's yes, 0 to 2 pi because there's no there's nothing really Interrupting the full rotation of the cylinders from 0 to 2 pi we don't have anything. Such as, like vertical planes. Okay, so now we have the boundaries and we have everything Ribbon in social hornets. Now let's calculate it. So first we're integrating with respects to Z. So let's do that. If we integrated with respective Z would have the data integral and the are into girl the same. And when we integrate with respect to get Z, we get our square times, E times, coastline data and the boundaries for ZR zero and our coastline data. Those are some data plus five and the art easy. Of course. DRD theta Sorry if we were to if you were to plug these boundaries into Z, we will get We will get the end to go The double integral Oh, though integral off R squared holds that data times are goes like you know those are assigned data. I was five PRT thing and we can distribute the are square coastline data into the arkels and edibles are sent data plus five to simplify it. So we get the double integral of of our cube cool Zendaya coastline square data, plus our cue science data time, schools and data. And if you notice, this is simply just an identity for sign of potato over two. So we can you can rewrite this as we can rewrite this, as are cute over two times. Sign of tooth data. And finally, five. Our square costar data D R. Do you think now we have everything in terms of our coastline and pretty simplified expanded form. So let's go ahead and integrate this with respect to our So, if we were to integrate this with respect to our it would be the integral, um, 0 to 2 pi Oh, one board. Since we're just gonna be using a bunch of power rules, 1/4 are to the fourth co science word. Whole science worth data Waas 1/8, 1/8. Sign when we were eight are to the fourth sign of to three. You know, plus 5/3 are to the third coastline data and the boundaries for our from 2 to 3. It's decided the data. Okay, so now, now, if we were to plug in the boundaries for into our we will get the integral from zero The two pi off. It would allow you again. 65/8. I'm sign of tooth data. We get 65/8 of sign to think so. Was 65/4 co sign? I swear it. If they don't, I was 25 over three. Close nine data. Do you think when the last thing we have to do now is integrate with respect to data? So if we were to do that, we would get you get negative. 65/16 coastline to think most like to theater plus 65 over eight. I'm stato plus something to think it over to this. You could get this into that theater goalpost and scored a that can be get by using Ah, behalf tingle the happening old formula for coastline data. And finally 95 over three. Sign off data and this is from to play from 0 to 2 pi. So if we were to plug in the interval into this equation, we would get a final answer of 65 pi over four

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 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

4 answers
Consider this reaction:NH, OH (aq) _ NH; (aq)- H,o (aq) certain temperature obeys - this rate law; rate (0.0980 V ')[NH,OH]"Suppose vessel contains NH, OH at _ concentration of 0.850 M: Calculate the concentration of NH OH in the vessel 45.0 seconds later: You may assume no other reaction is important:dlbRound your answer to significant digitsx
Consider this reaction: NH, OH (aq) _ NH; (aq)- H,o (aq) certain temperature obeys - this rate law; rate (0.0980 V ')[NH,OH]" Suppose vessel contains NH, OH at _ concentration of 0.850 M: Calculate the concentration of NH OH in the vessel 45.0 seconds later: You may assume no other reactio...
5 answers
Carbon tetrachorde ethyl alcohol bromine oil2) The solubility of ammonium chloride at 70. "C is 60. g of solute per 100. g of water: Which solution would be saturated at 70. "C? (2 points, show your calculation)a) 90. g of solute in 180. g of water b) 90 .g of solute in 200. g of water c) 30. g of solute in 50. g of water d) 35 g of solute in 50. g of water3) At which temperature would COz gas be most soluble?
carbon tetrachorde ethyl alcohol bromine oil 2) The solubility of ammonium chloride at 70. "C is 60. g of solute per 100. g of water: Which solution would be saturated at 70. "C? (2 points, show your calculation) a) 90. g of solute in 180. g of water b) 90 .g of solute in 200. g of water c...
5 answers
SID:Ipage) Page 10:1Sam-1.OSpm MTWThF 6/23/17 72 m #IB (ChI) [TOpts ] If ball is thrown into air with a velocity of 50 ft / $, its height 5 in feet seconds later given by 5_2- y = S0t - 16t2 Find the average velocity for the time period beginning when t =3 and lasting 0.5 25 seconds: 0.1 seconds, 0.05 seconds and 0.01 seconds. Estimate the instantaneous velocity when t = 3
SID: Ipage) Page 10:1Sam-1.OSpm MTWThF 6/23/17 72 m #IB (ChI) [TOpts ] If ball is thrown into air with a velocity of 50 ft / $, its height 5 in feet seconds later given by 5_2- y = S0t - 16t2 Find the average velocity for the time period beginning when t =3 and lasting 0.5 25 seconds: 0.1 seconds...
5 answers
31) How many valence electrons are present in an atom of vanadium? PLACE ANSWER HERE
31) How many valence electrons are present in an atom of vanadium? PLACE ANSWER HERE...
5 answers
4) Let A=be the augmented matrix for syslem equalionsDetermine for what values of. and the system inconsistentb) Detemmine for what values of = and the Bystem hag unlqua solulion:Determine for what values ol . and kina syaier ha: iniinilely many colutions.
4) Let A= be the augmented matrix for syslem equalions Determine for what values of. and the system inconsistent b) Detemmine for what values of = and the Bystem hag unlqua solulion: Determine for what values ol . and kina syaier ha: iniinilely many colutions....
5 answers
CHEM 1152L Lab Manual Page 61for structural isomers of the following compounds: Draw the structural Build models fonmulas and name cach; dimethylbenzenetrimethylbenzene$ine GTu.L"Jc a cunt 1') ` ^> -
CHEM 1152L Lab Manual Page 61 for structural isomers of the following compounds: Draw the structural Build models fonmulas and name cach; dimethylbenzene trimethylbenzene $ine GTu. L"Jc a cunt 1') ` ^> -...
5 answers
Heecmoite N JObieli19Par AThe exchange of oxygen and carbon dioxide in the lungs is a example of which law involving gases?Boyle's lawHenry' s lawAvogadro $ law Charles' lawSubmitRequest Answer
Heecmoite N JObieli19 Par A The exchange of oxygen and carbon dioxide in the lungs is a example of which law involving gases? Boyle's law Henry' s law Avogadro $ law Charles' law Submit Request Answer...
5 answers
Write a nuclear equation for the indicated decay of nuclide; Po-21O (alpha)Pb-21-(beta)0-15 (positron emission)Pd-103 (electron capture)Fill in the missing particles in each nuclear equation,AmJ Np
Write a nuclear equation for the indicated decay of nuclide; Po-21O (alpha) Pb-21-(beta) 0-15 (positron emission) Pd-103 (electron capture) Fill in the missing particles in each nuclear equation, Am J Np...
5 answers
Rz84SR1idealby er: pblerli al diflererczThz cirzuil #zer Jbove hzs beer corrlezled (or & ki0 lirlc rithTre %ilchcsilian Thz circuil prlairg lhrez ideticaliesis DIg ol (e-iglarC~GfaioiThe %ilc "ig ile cloge:Capgcilanzepotartial diffcrcnc- across tnc capactor #hc7(al Ir torrs 31 ieL: 4awit-h has t3c7opcn Ina -Irc" Justf tcur ;n8WCT .
Rz 84 S R1 idealby er: pblerli al diflerercz Thz cirzuil #zer Jbove hzs beer corrlezled (or & ki0 lirlc rithTre %ilch csilian Thz circuil prlairg lhrez ideticaliesis DIg ol (e-iglarC~ Gfaioi The %ilc "ig ile cloge: Capgcilanze potartial diffcrcnc- across tnc capactor #hc7 (al Ir torrs 31 ie...
5 answers
Find an equation of the plane passing through the points (1,3,2), (2,0,-1) and (-2,1,2)_~6x + 9y _ 11z = 1 ~6x + 9y + 82 = 1 4x + 9y + 82 = 1 4x + 9y - Ilz = 1
Find an equation of the plane passing through the points (1,3,2), (2,0,-1) and (-2,1,2)_ ~6x + 9y _ 11z = 1 ~6x + 9y + 82 = 1 4x + 9y + 82 = 1 4x + 9y - Ilz = 1...
1 answers
Find a rectangular equation for each curve and describe the curve. $$x=\sqrt{5} \sin t, y=\sqrt{3} \cos t ; \text { for } t \text { in }[0,2 \pi]$$
Find a rectangular equation for each curve and describe the curve. $$x=\sqrt{5} \sin t, y=\sqrt{3} \cos t ; \text { for } t \text { in }[0,2 \pi]$$...
5 answers
Question - Which 3 Wwl (2 polnts) of the follawing 1i Arthropod?
Question - Which 3 Wwl (2 polnts) of the follawing 1i Arthropod?...
5 answers
2 416,6fl ~C) 5 4aks EV Gtr)t owners believe will grow at 9% Quick Check worth 5100,00 But the for the business by finding the Assuming normal growth, a clothing store is reasonable selling price per year for the next three years Estimate a ~compounded semiannually: present value at 5% per year
2 416,6fl ~C) 5 4aks EV Gtr)t owners believe will grow at 9% Quick Check worth 5100,00 But the for the business by finding the Assuming normal growth, a clothing store is reasonable selling price per year for the next three years Estimate a ~compounded semiannually: present value at 5% per year...
5 answers
Causality True or false: If the sample data lead uS to conclude that there is sufficient evidence to support the claim ofa linear correlation between right and left arm measurements, then we could also conclude that increases in right arm measurements cause increases in left arm measurements_ Interpreting Scatterplot If the sample data were to result in the scatterplot shown here; what is the value of the linear correlation coefficient r?
Causality True or false: If the sample data lead uS to conclude that there is sufficient evidence to support the claim ofa linear correlation between right and left arm measurements, then we could also conclude that increases in right arm measurements cause increases in left arm measurements_ Interp...
5 answers
Polar equation r = 8sec(0) to a rectangular equation nm
polar equation r = 8sec(0) to a rectangular equation nm...
5 answers
Show that for an astic collision between m1 and m2, where the initial velocities of m1 and [ m2 are v1 and respectively; the ratio of the initial and final kinetic energy of m1 as function of angle given byMACOs8 =mi-m;sin"0
Show that for an astic collision between m1 and m2, where the initial velocities of m1 and [ m2 are v1 and respectively; the ratio of the initial and final kinetic energy of m1 as function of angle given by MACOs8 = mi-m;sin"0...

-- 0.020954--