5

Let R be the region bounded by y = 2Vr and y = I: Find the valune generated hy rolating the region R about the U-axis using Uhie waslcr method.(L) Fiud the vol gene...

Question

Let R be the region bounded by y = 2Vr and y = I: Find the valune generated hy rolating the region R about the U-axis using Uhie waslcr method.(L) Fiud the vol generaterl hy rotatiug the region R alont tho V-axis usiug the shcll mcthod.

Let R be the region bounded by y = 2Vr and y = I: Find the valune generated hy rolating the region R about the U-axis using Uhie waslcr method. (L) Fiud the vol generaterl hy rotatiug the region R alont tho V-axis usiug the shcll mcthod.



Answers

Sketch the region bounded by the surfaces $ z = \sqrt{x^2 + y^2} $ and $ x^2 + y^2 = 1 $ for $ 1 \le
z \le
2 $.

Okay, So working in the double integral, um, a region are the function e to the too wide divide by X Do I d. X. Given that the region is described by order bounded by these three functions y equals X squared y equals zero and y equals r Sorry, X equals two. Great. So the start off by drawing out what our lives like because then that will give us, like, an idea about how to write the limits of our double integral. All right, so people's too to slide this vertical line here. Michael zeros here. All right, for so it should be passing through here a perfect. And this is y equals X squared. Obviously, there's a lefty inside, but, um, that part doesn't really matter, because that's not what's being bounded. Um, but this little Kirby triangle right here is what I read in our looks. Think so. We can get a sense of what kind of limits we want to write for our double integral. So the double integral I right over here is a d Y DX intervals. So we want first, right? How Why is ranging. And then we want to write about how excess ranging. So for this Inter integral, uh, you see how Why ranges? And what I like to do is kind of draw imaginary line here and see, like passes through white was reversed. Always So are lower limit is zero for why, and then passes all the way to y equals X squared. So export is their upper limit. And then we have this these vertical lines passing through both of these. Why limits for all X equals zero to X equals two. So that would be our ex elements. Okay. And then this is basically the properly set up a double integral and start calculating and explicitly solving for him. So selling for the inner an inner inner girl. Um, just gonna rewrite this a little bit. So it makes it easier to see how we're gonna, um, integrate this because of those. A little tricky, But trust me, it's not. It's not too bad. So writing it this way, it helps show, like what the constants of our exponents are like the constant values of our exponents. Meaning are like basically in this case, would be to over x on. And why is r variable? Because we're integrating. Specialize. So every other variable that's not wise, seen as a constant in this case. So what we can do is we can use a little trick where, since we have basically a linear function inside of a exponential function Um, the trick I use is nor to integrate this I integrated as if you were regular like each the three x where it's of the actually, lemon, let me just show you what I do. So I integrated as usual, as if you were like you for the X, right? So I say each the two divide by X times y and then what you need to do is need account for this part. So because this is a linear function, all we need to do is just divide by that linear coefficient. So in this case, would be just to divide by X on. This would be, uh, settlements are users who x squared. And just to verify that this actually works, we can do is that basically we different or we integrate respect. Why so in order to see if, like we can get back to this, But we just needed to is partially drive it respect to why? Just make sure that it's correct. So by to about eggs, all right. And if we try to do this, it's basically you take this, um, e to the to divide by x times by everything stays the same because it's basically how you differentiate you, right? This is just a constant. So this comes along for the ride, and then you have to use the chain rule and we have to drive this part, and it should just be two over X. And then basically, this two over X canceled out with this two over X, leaving you with the original function I have. So just raise everything. So that's just verification that this little trick. So basically, if you ever have ah, linear function inside of another or complex function that you do know how to in a great you can use this like little division trip waits divide by the, um, Lear linear coefficient up done. And it should all work out. Right. So now just gonna be right this a little bit, um x over to each the two divide by X zero x squared. Um, and then we needed a plug in our limits since we integrated respect to why we're gonna be playing these limits into why not X? So just be careful of that X e to the to divide the X plug in X squared bye bye to minus what we get when we plug in zero So x each to divide by X times zero bye bye to this whole thing should become one because zero times any number is still zero and then e to any number hurt e r any number races or of power is always one. So one times X over two is just sort of So we have X e This should just some fight to this. Bye bye to on that minus X divide right to So this is what we get from the inner limit or the inner integral. And now we need a please start on the curb place in the outside integral. So, sir, to two of this quantity X b to the two X survive by two minus X over to d X. Um Okay, so what I'm gonna do first is gonna just split the interval as so minus two. Two d x and this just allows me to, like, separate out. Um, I looked at this part, and it looks like we need to use integration by parts. So just to take this out of the way, I'm just splitting up the interval. So then I can work on this piece separately and this piece of relief and not having them combined together. Um Okay, so let's tackle the harder part. So the integration by parts, Um, I'm sure you guys know about the acronym lying where you set you to be, whatever it comes for. So l for logs, I for inverse trig A for algebraic or basically a polynomial, uh, t for trick and e for exponential. So basically, you set you in this priority. So you had sent you to either a log first. If you see a log here, uh, you to be a inverse trick if there is inverse trick and then so on. So forth. Um, this case, I see that there's an aggravated component x over to first. So let's just set you to be we're after Is this that you to be x and then the other company would be devi devious, set to everything else. That's not included. So Devi, which is B E to two X, abide by two DX. Let me just make sure you too. Yep. Okay. And then reason you find to you, do you, um we were just different shades. Respect to X On the right hand side, we just get DX and then we integrate for the Devi part, So V is equal to we integrate this part, and it would just be e to two x divided by two. So I'm just in agreeing as if it were a regular e to the X and then we just need to divide by that, um, linear coefficient inside some using the truth I used before And that would be V. And this would just simplified seats that two x two bye bye for Okay, So once we have all this, we know that the integral of yeah you d v is equal. Teoh u times v my integral from are integral of e d u. Okay, so applying it to this integral we get is the integral of 0 to 2 of x e 22 x divide by two DX since you is just X and V is part devi is this whole thing. It means that, um, it's equal to you. Tens be so x Times eats the two x divided by four from zero to minus. Then grow from zero to a V D u So v do you ve is this to eat the two X divide by four and d you is just steps, okay. And then we just need to sulphur this entire thing here. So this is gonna be equal. Teoh to e to the fourth over. Four minus zero. I just clicked into top limit first and then the bottom limit second. And then when you plug in the bottom limit, you get zero because yes, would become zero. And any number of times zero is zero minus eat the two x divided by eight again. I'm just using that Lanier trick. I just divided or I brought e 22 x over four first and then I divide by two. And then I did the all in one step, and then this is evaluated from zero to. This will be equal to to eat 1/4 over four minus. We get when we plug in to so each the fourth over eight minus what we get when we plugged in zero, which is just 1/8 this is equal to you. Said for over to each of the fourth over eight. Plus one teeth. Right. You think? Let me just make sure. Um yeah, okay. And then this would just simplify to four years the fourth over eight, East, fourth over eight and then lost 20. So this would result in three E to the fourth plus one over a. Okay, so we evaluated this integral. And now we just need todo with this simple and a girl pretty easy. Um, we do that here. So negative of the quantity X squared over four from our reverse power rule here, Um, from 0 to 2, there should be able to and plug into 4/4 minus zero. So you get a negative one here, So basically, we just need a subtract one from this result. So now we have, and that should just result in Let me just simplify a little bit more, so there should just be three each of the fourth, minus 7/8. Andi, I think that is our final answer. Yeah, that is our final answer.

So here in this question we have to sketch the parable lights Which is z equals two x square. Bless the white square and Is he equals 2? Two minus x squared minus y square. Now I'm going to plot this parable Lloyd 1st on this graph paper. Now it looks like this, it will be an upward para polaroid like this and it will pass from Z equals to zero, this is equal to zero. And here this is one, this is two. The next one, this is two minus X squared minus y squared. Now this would be a downward parabola Lloyd And it will pass from z equals to two like this. No, the region that we need to sketch is this region. This is basically this would be a 3D region. So this is the solution that was asking the question. I hope it was a problem. Thank you.

So the question asked, You find a volume under which in that he's above find verbal light on below by the chang er in the X y plan. So, first of all, we need to turn uh, region in the X Y plan to set up the limit under the intercourse. So change. Try it now. Thanks. And why here? So why would you think so? We need to have this. I ain't going through the regime this like being wanted. Could you axe? And we have ex coaches are Oh, so this will be another night? Yeah. Next big good wages rose here. So thanks. You could user. And another life x plus y You could teach you So listen, you can write it down as grind. We could you two minus x. So it means that we have the Jews here and ah, went acting then we haven't under tonight. Yeah, bring down. Okay, so the to lie this life will be why you go juju minus thanks and end to find us in this section. Here we have Juliette. Why did you do a humanist accent? Echo Jew. Thanks. So, Amy and Snap do we go to your XO XO ego to one. So we have this money to get you one. And then the father Why did you want this? Well, okay, so the region then we're looking for would be in this Django here, One kill and three. So this will be in the region in the X Y plan. So the volume can be computed by the pool in the girl and now Ana intercom inside will be their function here. So we'll be the X square plus. Why square? So now we need Thio. Uh, either that the extra Guan otherwise the ex. So we see that it would be much easier to do the d Y fast. So do I. And the act. So the wine we can go from the bottom here on which the top so the bottom correspond to India. Thanks, Onda. Tough response. You'd, uh humanist. Thanks Onda makes them the ex girl from the left to the right girls from Inman Value which is zero to the maximum. Well, no, just one. So we'll have ah formulate the double into coach Been the volume Aborigine now. So what? You need to go hatch to even just integral here. So we will keep it out into column from 0 to 1. And now from inside we have Why is a parable? So the X is a constant. So we should have the X Square Times, Kwai And now, plus know why Squared interview on the West Coast, actually. Wind power tree your family tree. So this one would be to limit from extruded to minus X the Yanks and there is an equal Jew. We didn't find it inside. So have from that one again so we wouldn't Jew minus X in. And then we shouldn't have the X Square to minus thanks on glass to minus. Thanks. About tree. Never bed three and then we subject No, we were the X in. So we have X about tree and now a minus. Thanks. Power three, remember? Three. Thanks. So we need you soon fight one more time. So you could you go from that one. So we went inside for the first time. We should have that X square minus expert tree class. This one was again, huh? And we expand everything. Which, again that, uh, on the question and expense I would give ju minus next about tree for now, given by a tree and then we mind us expound tree minus. Thanks about Jeannie, Remember? Three The Yanks. So it isn't a quit you integral to one. So now we have that exclaimed Men, do the extra first, uh, have minus one extra and management under one x tree and then minus thought x tree so we shouldn't have the you Children's TV 67 So Mina's oven number three expound three and then plus X square And then plus Jew minus Hank square. Now, uh, axe about to end of a big tree. Dignity x So this one we can do ah into go one by one the first time we have reread a constant months tree It's been with three times. Expert trees are integral expert You will be expelled for you before class That second time X squared So integral Exquisite beings luxury Remember tree So this one will give rooms that you won Now the last term we can't wait for us. We were I It would be one of the tree into go from that to one and in some have two minutes. Thanks. Now a three D Thanks. Now for this one here. We can use the change variable. So you could you true minus X so d'you equal to minus the angst and when acts it could use. Oh, is it implies you you could do ju. And when Thanks. Go to one. Then you equal what you will. But the one inside and we'll have one. Okay, so this one will be reflecting on. Re rented is on. We would have won in Jude off limit. Here's gonna minus three Tim's phone. And now it does one on the tree when was over. Museums. And that's fine. Get us one of the tree. So now in the limit goes from 2 to 1 and now it inside we have you our tree and the ex secret humanity You So this one could be minus. Yeah, you. So we have is an ICO. June minus seven. Number three. So, money seven come but drowned. Plus So that's a confection with Thames Far. Everything's kind of fallen on with a round. And now my nose, one on the tree. So you about read into comes about through me. You performed even before on just a minute. Go from 2 to 1. So we parted on ico Jew. If the 1st 2 affection you could you minus seven rest home will be three minors. One moment, three No, we have We will limit incisor without of one off on minors to from afar with a 16 Have fun So we didn't find his one Who should get Nico So 3 November it drowns We're gonna one on the far runners one on the three. Uh And then we have This wouldn't be one off minus fart as extending every father Could you fall so everything should be Could you one off minus one on the trail Plus for on the tree So we were everything in the denominator around So they're gonna first infraction will give us a good three minus one plus with hams for him will be 16 So we should get everything home with drown So three bless extinguish Between night Dean on man minus wouldn't be 18. So isn't will Echo Ju. We remember a six to a gun to a tree. I'm with Jew and that's a friend Answer

Herring. This problem we have to find the volume for the region bounded by the parable. Lloyd Z is equal to X squared plus y square on the triangle enclosed by the line y equals two x X equal to zero on X plus y equals two to the festival. I'll be graphing these equation on a graph favor to find a limit for the integration. So let's say a graph ever know. So this is the Y axis on. This one is x X is not. This line represents the equation explicit, like world strip, too. On this one is wise equals toe X on this one is X equal to zero on the boneyard region by the triangle is dis region. So the volume would be quilts to the Dublin Trickle R d sorry, X squared plus y squared d that will be equal to the volume where r is the region bordered by the this triangle. So we had right the limits and we know that d a sickle Rudy extra at the x dy dx dy y So the Dublin trigger X squared plus y square dx dy way Now to write the limits, we have to split this area to do part this part in this spot. So this is a part one on this one is the part two. So I'll be taking hold in the limits. Hardy handles this cross section. So here, x zero. So you get the limit for X zero on this side. Excess golds two tu minus way. So I'll get to minus way now the limits for a div. I would very between this line on this line hear why he's one on here. Ways to bless the double integral X squared plus y square dx dy way. Now I will be writing the limits for this area. So here x is you know, on here X is a why on the limit for divide would be between this line and this line of hair y zero on here. Why goals to one to finally we have got the volume We're just simply by these two and wriggle to find the volume of the required region So let this integral be as I won and let this integral bs I toe So first of all I'll be solving I want and then resolving I to so Isaac worlds to the Dublin trigger. Zero to minus way. Want to do X squared plus y square dx dy way. First of all, I'll be solving the inner integral first so it will, because two ventricle oneto on her good x cubed, divided by three plus y square x on the limits are from zero to minus y. Doctor doing now simply lb Plugging the limits So want to do I want to do on Dhere? We get a to minus way. Divide by three on a power destry Bless y square. Here it is two minus way. Not doing now will be a simply finder the inner function So you get ventricle one toe to here it is two minus y to the power. Three. Divide by three. Blessed to White Square minus y cube. Daughter die. Now I'll be integral. I'll indicating the inter function. You could say that this dysfunction so we'll get to minus y to the power. Four. Divide by four times three's 12 on one negative. Sane this two y cube divide by three minus. Right About four Divide were full on the limits are from 1 to 2. Are simply going to plug elements for the parliament and in the lower limit. So we'll get the value off Ivan as one now will be. Simply find the second Integon that is I to on It was a cool shoe double integral 02 away, zero toe one x squared, plus y square dx do a So here I'll be sold in the analytical first, So I get 0 to 1 on X cubed divide by three minus y squared X on the limiter from zero to Y daughter delay that will be quelled student and wriggled 0 to 1. I've been plugging the limit stone, so why cube? Divide by three minus y cube on the limits. Three. Dr. Delay off plug limits and regard this expression. Now I'll be integrating. So I'll get Dwight with about four divide before times three stools minus voted about four Wherefore on the literature from 0 to 1. So after plugging the limits, really get the value s one by three. So finally volume is equals Toe I one plus I to and I won was one and I do is won by a tree. The result would be four by three. So finally we can compute Dad. The volume for the desired region is four by three to the final answer


Similar Solved Questions

5 answers
Solve sin? (t)6 cos(t) for all solutions 0 < t < 2T ,Give your answers a5 values accurate to at least two decimal places in a list separated by commas_
Solve sin? (t) 6 cos(t) for all solutions 0 < t < 2T , Give your answers a5 values accurate to at least two decimal places in a list separated by commas_...
5 answers
Prove that there is no exacl identity" for convolutions in the sense that there is no absolutely integrable function 0 R such that 0 f = f for every f € BC(R)- Hint: Argue by contradiction_ Assume 0 is an exacl identity' choose (fax-1 t0 be an approximate identity (satisfying some additional properties that you will need t0 determine), and look at the sequence ol numbers 0 * f,(O))xx-1. (Caution: 0 might not be continuous at € 0.)
Prove that there is no exacl identity" for convolutions in the sense that there is no absolutely integrable function 0 R such that 0 f = f for every f € BC(R)- Hint: Argue by contradiction_ Assume 0 is an exacl identity' choose (fax-1 t0 be an approximate identity (satisfying some ad...
5 answers
Sin T .cos(a + z)+ tan cos 1. Simplify: cos sin( 2-a)+
sin T .cos(a + z)+ tan cos 1. Simplify: cos sin( 2-a)+...
5 answers
Volume of 2.8 L of air at 37*C is expelled from the lungs into cold surroundings at 6.89C. What volume (in L) does the expelled air occupy at this temperature? The pressure and the number of gas particles do not change.
volume of 2.8 L of air at 37*C is expelled from the lungs into cold surroundings at 6.89C. What volume (in L) does the expelled air occupy at this temperature? The pressure and the number of gas particles do not change....
5 answers
To convert fr}m degrces Celsius degrees Fahrenheic: Aateee the forul f(z) bc sun You could explin ic mexningFind the inverse function; if it cxists a1E;f" '(s) =help (formulis)
To convert fr}m degrces Celsius degrees Fahrenheic: Aateee the forul f(z) bc sun You could explin ic mexning Find the inverse function; if it cxists a1E; f" '(s) = help (formulis)...
5 answers
Asagidaki fonksiyonlarin Ianim bolgelerini bulunuzcizimlerini yapiniz.fonksivonlarin Yanlarinda verilen noktalardaki vursa limitlerini hesaplayiniz ve surekliligini ufuSlifinz +X-> s+) 4) 2 (x Jo) = (1.2) D) = (Jo)- (0.0) ~1+3x-y
Asagidaki fonksiyonlarin Ianim bolgelerini bulunuz cizimlerini yapiniz. fonksivonlarin Yanlarinda verilen noktalardaki vursa limitlerini hesaplayiniz ve surekliligini ufuSlifinz +X-> s+) 4) 2 (x Jo) = (1.2) D) = (Jo)- (0.0) ~1+3x-y...
5 answers
Find the basic eigenvectors of A corresponding to the eigenvalue A.0 -1A=1 = 3-8-3Number of Vectors:
Find the basic eigenvectors of A corresponding to the eigenvalue A. 0 -1 A= 1 = 3 -8 -3 Number of Vectors:...
5 answers
Cousiderrod of length ad MTASS which is pivoted at one end. point mass m is attached to the free end of the rod,1. [ = L?2 None of thlese.3. [ = m L?4 [ = m L?5. [ = m L? 413 6. [ = m L? 12Determine the moment of inertia of the system with respect to the pivot point. The ncceleration of gravity g = 9.8 m/s?7.[=2mD
Cousider rod of length ad MTASS which is pivoted at one end. point mass m is attached to the free end of the rod, 1. [ = L? 2 None of thlese. 3. [ = m L? 4 [ = m L? 5. [ = m L? 4 13 6. [ = m L? 12 Determine the moment of inertia of the system with respect to the pivot point. The ncceleration of gr...
4 answers
Which sets of numbers are Pythagorean triples?$$6,8,10$$
Which sets of numbers are Pythagorean triples? $$6,8,10$$...
5 answers
Let fhe wice-differeuttable function sch 0 forall x The graph of y = S(x) Is thc secunt line _ passing through the points (3,f (3)) and (5,f(5)). Te gTph Tlx) Is,lhe linc tangent ! the grph of f m Which of the following trua? f(42) < 5(+.2) T(4.2) QBL#ARTI <JTZ) 38421 <(42) T(42). (DLILL[LL S4LLL
Let fhe wice-differeuttable function sch 0 forall x The graph of y = S(x) Is thc secunt line _ passing through the points (3,f (3)) and (5,f(5)). Te gTph Tlx) Is,lhe linc tangent ! the grph of f m Which of the following trua? f(42) < 5(+.2) T(4.2) QBL#ARTI <JTZ) 38421 <(42) T(42). (DLILL[LL...
1 answers
Changing temperature along a circle Is there a direction u in which the rate of change of the temperature function $T(x, y, z)=$ $2 x y-y z$ (temperature in degrees Celsius, distance in feet) at $P(1,-1,1)$ is $-3^{\circ} \mathrm{C} / \mathrm{ft}$ ? Give reasons for your answer.
Changing temperature along a circle Is there a direction u in which the rate of change of the temperature function $T(x, y, z)=$ $2 x y-y z$ (temperature in degrees Celsius, distance in feet) at $P(1,-1,1)$ is $-3^{\circ} \mathrm{C} / \mathrm{ft}$ ? Give reasons for your answer....
5 answers
Tne hiquid ethyl propanoate his density of 0.884 EInL at 25.4 "€ Malouu 25. Intemnst provided'cnmnnundptovided,Of thc Iiquid aLutniianowerHeby Entre orolaWUce unomhulutnple emainm
Tne hiquid ethyl propanoate his density of 0.884 EInL at 25.4 "€ Malouu 25. Intemnst provided' cnmnnund ptovided, Of thc Iiquid aL utniianower Heby Entre orola WUce unomhulutnple emainm...
5 answers
Homework: HW 7.4 (18) ScomDiollmDul T8f COpa "FScon 79528 Ranio An7.4.69IozfnnU InfIOS= ean37 ninti FuFtelhil FSortt Fu [7 Ete Aetae DneceCmtnEateLMeODHuntmen
Homework: HW 7.4 (18) ScomDiollm Dul T8f COpa "F Scon 79528 Ranio An 7.4.69 Iozfnn U InfIOS= ean 37 ninti Fu Ftelhil FSortt Fu [7 Ete Aetae Dnece CmtnEate LMe OD Huntmen...
5 answers
0usin3;4ptsHow Much force riust be applied snnE, #hich nax How much cnergy will storco chut: * pi ing?NrAL Gt 120 N/n, to stretchit 150J80N,135JBDJN 13500 ]JaN 135 |en135 tekcu
0usin3; 4pts How Much force riust be applied snnE, #hich nax How much cnergy will storco chut: * pi ing? NrAL Gt 120 N/n, to stretchit 150 J80N,135 JBDJN 13500 ] JaN 135 | en135 tekcu...
5 answers
Find a parametrization for the curve_The lower half of the parabola x+8=y2 Choose the correct answer below:0 A xsty=t +8,ts0 0 B xsty= ? _ 8,t28 0 c. x=t?-8,ytt20 0 D. x=p+8,y-tts8 0 E. x=?-8,y=ttso 0 R xsty=t +8,t<8
Find a parametrization for the curve_ The lower half of the parabola x+8=y2 Choose the correct answer below: 0 A xsty=t +8,ts0 0 B xsty= ? _ 8,t28 0 c. x=t?-8,ytt20 0 D. x=p+8,y-tts8 0 E. x=?-8,y=ttso 0 R xsty=t +8,t<8...
5 answers
The figure below shows the graph of_ Fatona function f: It has verticel asymptotes x = and horizonta asymptote The graph has x-intercept and passes through the point (-3.~1)The equation for f (x) has one the five forms shown below: Choose the apprapriate form for f (x) and then write the equation. You can assume that f (+) slmplest form .0 f()=8 0 f() 26-0 E8 {6) 06 J() = 0 (T 38 c6 - a) r6) = 9 363 D6 -o 4)( (+26 +0
The figure below shows the graph of_ Fatona function f: It has verticel asymptotes x = and horizonta asymptote The graph has x-intercept and passes through the point (-3.~1) The equation for f (x) has one the five forms shown below: Choose the apprapriate form for f (x) and then write the equation. ...

-- 0.021810--