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Evaluate the integral4rzdSwhen S is the part of the planeT + 29 + 2-shown inenclosed by the cylinder2+y? = 4....

Question

Evaluate the integral4rzdSwhen S is the part of the planeT + 29 + 2-shown inenclosed by the cylinder2+y? = 4.

Evaluate the integral 4rzdS when S is the part of the plane T + 29 + 2- shown in enclosed by the cylinder 2+y? = 4.



Answers

Evaluate the surface integral.
$$\begin{array}{l}{\int_{S} y d S,} \\ {S \text { is the part of the paraboloid } y=x^{2}+z^{2} \text { that lies inside }} \\ {\text { the cylinder } x^{2}+z^{2}=4}\end{array}$$

Okay, Apologize for the delay. I was having trouble with my pen, and I'm hoping that I could get that back at this point. Um, this in a girl is going to require your u substitution or u substitution is used whenever my portion of my function. Um, that fits a pattern that is similar to something we've seen before. This will be my you and then the derivative of three tea. Since it's a three times, do you? Then it just differs by a constant. So I'm gonna look for that, do you? Over you substitution. So I'm gonna start out by coming up with what my substitution values should be. I'm gonna start with my you is the denominator three t plus four and then my derivative of you or d'you differential three times d. T landing me on a substitution for DT, which will be d'you divided by three. That's gonna allow me to go ahead and come in here and change my d t into do you over three and my three t plus for into it you. But that's not enough for me right now, because this is this is a definite integral and I'm gonna want to change my limits as well. That's just my own personal preference. I prefer to change my limits so that I don't have toe return to the T values before I wait. So remember that if you is three times t plus four, then in the case that tea is equaled it too. Three times two is six plus four is 10. So my you limit converts to attend and in the case of tea is 12 three times 12 is 36 plus four is 40. So my you limit changes to a 40. Now that is ready for me. Substitute and evaluate. So at this point, what I'll be able to dio is rewrite my integral entirely in terms of you values. So my lower limit is now a 10. My upper limit is a 40. My function, you and the denominator. And that and the numerator is my d'you over three. Well, that's a little messy. I don't want the complex fraction, so I'm gonna take the 1/3 and put it out front and, um, put that in red. So I remember it's there. 1/3 time's the anti derivative from 10 to 40 of D over you. Now look at how nice and clean that is. Fits my rule to a T. So the anti derivative of do You over You is natural log of absolute value of you. I don't need the plus C because this is definite. And don't worry, I didn't forget my 1/3. I'm gonna go back and put that in right here. Last time I did this, it changed my pen thickness, and that was frustrating. Okay, now 40 is gonna go into the absolute value as well as 10 neither of which is a positive number. So when I evaluate this, I'm not gonna need my absolute value. So all I need here is natural log of 40 minus natural love of 10. So here's what happens at this point. At this point, I'm noticing that I have one of my natural log rules at play and to review that the natural log roll that's applying here is natural. Log of a minus natural log of B is the same as natural log of a divided by B. So this is equivalent to natural log of 40 divided by 10 or natural log of four and don't forget, I'm gonna carry down by 1/3 and multiply lots of different ways. I could write this using my laws of logarithms, but I think we'll just stick with 1/3 natural of four or you could say natural long for divided by three.

Here. We're looking at the region in the first walked in. So that's where everything's positive. This is it right here and then Not only in the first locked in but bound by this cylinder and this plane, Why equals three. So we're looking to right six iterations of the triple integral right. Which is how we're going to sell for volume. So volume standard is up This form where D V is some iteration off DZ de why d x So then switch around the order of easy to hide. The X is where we get that three iterations, and it changes a little bit the way that the bounds of these intra girls is written. But I'm gonna start with d walk DZ d Y d x cause that's kind of Ah, a standard format. Do you see? Do you buy T x? Okay, so we start with easy and we take our equation and we sold frizzy pretty straightforward C Z squared equals four minus X squared. So Z equals square root of four minus x squared 34 So that's our upper bound. Normally we would say Z equals plus or minus, but this is founded in the first octomom slow around 20 Then the next is why. But we have this. Why equals three? Let's straightforward. It gives us an upper bound of three lower ground zero because of that 1st October. And then finally we have X. So we take this equation we were working with before set Z equal to zero and then we're gonna solve for X zero equals the square root of four minus x squared square Both sides We get four equals X squared, right Adds X squared both sides. So then X equals plus or minus two. So our upper bound for the X integrations to normally the lower bound would be negative too. But again were bounded by the first Acton's with zero. This allows us to move on to a second generation that also has DZ first, the web T X T y. Meaning our first upper and lower bounds here will not change. And then the same work applies for ex integration and our why integration as well. We just change the order a little bit appreciate for it. Okay, so now we have to change things up a little bit because Deasy is no longer. First, we've done the two iterations where that's the case. So now we're gonna do D X. Do you? Why d c so taking our equation? We're in a solve for X. So X squared equals four minus C squared. So X equals square right before minus c squared and again like before. This would normally be plus or minus, but because we are about ended by the first awkward tucked in our lower bound is Europe. And then why, once again, straightforward 0 to 3. And lastly. So we're set X equal to zero and so frizzy square. Both sides four minus c squared equals zero c equals plus or minus two. Her upper bound is to lower bound is zero instead of negative to again because of that found. All right, so then this enables us to continue on. We're gonna keep that d x in the front. Gonna have easy, do you? Why? So this should look familiar Because again, we're just changing the order of those last two bounds. Because why is separate? It's not impacting the way we sell face. That was simple. And then lastly we begin with Do you want d. C t X So when we begin with d Y, it's the same process 0 to 3. And then we know what happens when DZ is here. Make more than that because when we set the equation equal to Z, we got square root, a four minus X quipped, And then lastly, t x from our previous problems. We know that 0 to 2 all right. And that our final one again starting with D Y in the way of d x d C 0 to 3 and then looking at our previous iterations, we confined four minus z squared 20 and then Syria's two and these are six iterations. However, we're still not done because we need to evaluate one of these. Okay, so the one I'm willing to evaluate here is are DZ dx dy by So if zero to the square root four minus x squared and then you zero to and 0 to 3 get ready because we will be using a lot of Trigana metric identities here. Okay, so beginning and diving in we have Z and then we substitute thes two values, right? All of that. Txt. Why? So we can substitute that in I'm gonna write a little smaller so that I don't know how to read, because we're gonna have a lot of steps here in the next few minutes. Square root four minus x squared. T X D line. All right, So here's where we come into our first substitution. I'm gonna substitute X equals to sign, have you? And then with that, we end up with D of X equals to co sign you, Do you? All right, so I could make this substitution. Now get 0 to 3, and I'm gonna leave these bounds alone because we're gonna actually substitute those. So if the square root of four minus to sign you, it's weird multiplied by two co sign you. Do you do what? And then plugging in our initial X bounds zero to into this substitution, we get zero and high, divided by two. Okay, so completing. I want toe address this experiment. So 0 to 3. Where to? And then we get four minus for signs square you and then the rest of it stayed the same. And then I'm gonna factor out my constance and bring those in front of my integral someone a factor out a two out of this square root in this to as well to get four. And I'm just getting out of that that out of my way for simplicity sake. You really don't want to deal with any more information than we need to. We haven't tough already. City this and then there here, you'll note you're gonna metric identity. This equals one minus sine square U equals co sign square. You allowing us a simple by this A little bit, right? So then the square root of co sign square to you is co sign you multiplied by co sign you. Do you do what continuing. We want to combine like terms. So then here we can rewrite coastline squared you as one plus co sign to you divided by two and then I'm gonna multiply by two right to get rid of the fraction here I'm left with two out in front and left with one plus co sign to you. All of that. Do your deep water. Okay, now we can address the integration. It's too no final Integral Still left you plus 1/2 sign to you. Oh, that from zero two pi twice by two do you want? All right, Now we can substitute the hi divided by two in and I'm gonna go ahead and multiplied by two as I go along. So I end up with hi, right? Plus zero because sign of pi zero and sign of zero is also zero. All of this do you, and moving on to our final steps We end up with hi. Why? From 0 to 3. So three pie, my zero equals three hut. And this is the volume of our region. And it's the same volume you would get whether no matter which of the federation's, you would evaluate the problem.

So for this problem are going to be one. He's in the formula, that thes surface integral of f x y z the US is equal to the integral of f of our UV quantified by the magnitude of the cross product of you envy what do you deviant? And so we were given precision that why squared plus Z squared is equal one. This will be equal to a cylinder, um, with an X axis at the axis with radius of one. So if you want a parametric eyes this, um, this will give us the co sign you assigned you and since s is the part of the cylinder is in the first auctions X and Y both have to be positive. That's only going to happen when you was between zero and pi over. So it's just us circle. And since V we can see right here, a V is equal to acts and we know that our bounds both ex being equals zero and X equal to three tells us that the bounds for V will be zero and three. Now that we have all that information set down, we could start working towards solving the problem. First thing we need to do is come up with the magnitude of the cross product of unity. To do this will need both EU partial and V partial, but says R is equal to V I plus co sign you of J plus sign You okay? The u partial are you will be equal to zero. I minus. Sign you, Jay. Let's co sign you. Okay. And then the V a partial will be equal to I plus zero j plus zero k as neither of the J or K components have a bee in them. So I only cross these two together. Are you cross RV? This will be equal. Teoh I j ok of zero negative sign you co sign you of 100 and this will be equal to I times Negative sign you a co sign you up saying you 00 minus did a shay component zero assigned you 10 plus k time zero negative sign you 10 and then solving this will give that the cross bonnets of the U. N. V Partial is equal to zero. I plus co sign you, Jay. Let's signed you okay. And so the magnitude of the cross product be able to the square roots, the cross product squared. So is your escort plus co sine squared plus sine squared in Khowst and script of science crew is equal to one. So the square root of one we'll give that the main suit across parts is equal. One. It's now plugging into our ah formula from earlier. Uh, we know that the surface integral of Z plus X squared Y t s. We can use her parents ization and substitute out z x and y In this case that will give out sign U plus v squared because excess square multiplied by co sign of you and I will be almost supplied by one you d. V Now from here. Once you plug in, we'll have that. This is equal. We'll have that. This is equal 2030 to pi over two. A sign U plus v squared co sign you. Do you devi And this is integral Weaken solve without having to do any substitution or similar tricks to solve this problem. And this will give ounce that this is equal to the angle from 03 of V squared plus one DV, which is equal to the cubes over three plus V from 0 to 3, and that will be equal to three cubes over three and plus three. Three cubed is 27 27 or three is nine so nine plus three until it is that the service area were integral is 12.

I have to evolve it. Is this in temple sign d t October or plus he square and limit goes to Do you want to the value of this? We have to substitute d bye to 10 ft. Oh, differentiate it, DT Equals to who? His exc wired theater into the theater. We substitute this. We they decided the limits for make it easier. So DT equals to two six square into the free to Yeah. Thus for Entity Square, which is for hand squared detail. Okay, no, we can write. Eight, 26 10 the data into four into one plus 10 square feet, which is 86 square today. So we can right to 6. 30 later, dictator into to say three down now it will be eliminated. And finally it will result in and people of 60 to in terms of the birth of the theater. So we can easily integrate this, Which is long Ellen Stick detail, huh? 10 to Mm hmm. No, we aren't coming up from the values of Tika. We considered a right angled triangle. This is so we assumed he equals 2 to 10 liters of 10. Peter, he was two t by two. So this is T. This is to earn this becomes fruity square us for so from here we can write l m 10 today calls to t v very ready to arise To get the full story. It tastes square plus four invited went to now I put the limits on which is 0 to 2 Yeah, it proved their values and key. Yes. 22 to minus Ln school to us two by two Mhm. Finally, results in l m were to first one. So this is the antique Well.


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