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Express the integral f(x, Y z)dV as an iterated integral in six different ways, where Eis the solid bounded by 2 = 0,x = 0,2 = y - Sx and y = 20.82(1)hz(*,J) f(x,Y,...

Question

Express the integral f(x, Y z)dV as an iterated integral in six different ways, where Eis the solid bounded by 2 = 0,x = 0,2 = y - Sx and y = 20.82(1)hz(*,J) f(x,Y, 2)dzdydx (xJ)1h (x,82(y)hz(*,) f(x, Y, z)dzdxdy (xy)g1(y)hi(x,h2(x,82(2)h2(,2) f(x, Y, 2)dxdydz h Cy,2)8 (2)81(2) hi(y;hz(y, 2)32(y)h6,) f(x, Y, z)dxdzdy C,2)81(y)g1(y) hi(y, 2)hz(y, 2)82(r) h2(x,2) f(x, Y, z)dydzdx 81(x) h (x,2)h1(x,hz(x,82(2)h2(x,2) f(x,Y, z)dydxdz h (x,2)8(2)hi(x,hz(x,

Express the integral f(x, Y z)dV as an iterated integral in six different ways, where Eis the solid bounded by 2 = 0,x = 0,2 = y - Sx and y = 20. 82(1) hz(*,J) f(x,Y, 2)dzdydx (xJ) 1 h (x, 82(y) hz(*,) f(x, Y, z)dzdxdy (xy) g1(y) hi(x, h2(x, 82(2) h2(,2) f(x, Y, 2)dxdydz h Cy,2) 8 (2) 81(2) hi(y; hz(y, 2) 32(y) h6,) f(x, Y, z)dxdzdy C,2) 81(y) g1(y) hi(y, 2) hz(y, 2) 82(r) h2(x,2) f(x, Y, z)dydzdx 81(x) h (x,2) h1(x, hz(x, 82(2) h2(x,2) f(x,Y, z)dydxdz h (x,2) 8(2) hi(x, hz(x,



Answers

Express the integral $\iint_{E} f(x, y, z) d V$ as an iterated integral in six different ways, where $E$ is the solid bounded by the given surfaces.
$$x=2, \quad y=2, \quad z=0, \quad x+y-2 z=2$$

All right, So here's another long problem. Won't express this in six different ways. So first thing we do is we assume that we're just dealing the X Y play. Why? And we find are our bounds like this. Do you know that Z is bounded below by zero and bounded above by? Well, whatever this is when we saw for that's all for Z, right? So you set up that integral. So notice that our disease, we're not going to be changing within these two in today, You know, I think it's going to be changing. Is depending go on the region. One region to region one. Yes, one. This guy is region too. Yeah, this is very straightforward. Uh, and then we do the same thing for the y Z plates. X equals zero. All these guys here. Fantastic. So then we solved way. Figure out the bounds bounce Drunks are going to depend on again This first guy here bounce one bounce crazy, then region one versus region too. Got our bound like that. What? Who's straight forward there? Okay. And, uh, So what do we do next? Well, we do the X z plane and again very, very similar same time or why rounds don't change but our Z Valens and expounds depending on Region one versus two.

For this problem. We need to evaluate a given triple, integral. Now, anytime you have something like this, it looks very complicated. And the first question is always Where do I even start this? Well, if you think back to algebra, if you have nested parentheses, you always go inside. You go into the innermost set of parentheses. Evaluate that and work your way back out. The same holds true for integral. We're going to go to the innermost integral, evaluate that and then work our way out one at a time. So we'll evaluate with respect to why? Then we'll go to do the one with respect to X, and we'll finish with the Z. So let's take a look at what that innermost piece looks like. These pieces here are just gonna leave. But what about this, uh, innermost? Integral? If I'm doing this with respect to why that means, why is my variable everything else I'm going to treat like a constant. So in this case, I have each the negative. Why? E is kind of his own. Integral. All I need to do is have a negative so negative x each of the negative. Why would be the integral of that innermost piece. And I'm gonna evaluate this from y equals zero to the natural log of Z. Okay? Everything else. I'm just gonna leave out there for the moment. Okay? So what do I get? Well, if I go to my upper limit, why equaling the natural log of Z E to the negative natural log of Z is one over Z So I'm gonna end up with negative X over Z. If I let y be zero, each of the zero is one. So I'll be subtracting a negative X. Yep, which is equivalent to having a plus X. Okay, so there's that. First I have I have successfully evaluated and dealt with that innermost, um in a girl. Now let's do our next innermost. That's the one with respect, Toe X. So I'll leave the outer peace alone. But what do we get when we evaluate that that innermost integral. But now we're doing this with respect to X. So Z is going to be a constant X is my variable. So I can just use my, um you know what? I need to go back. This limit of integration is to Z it's hard to read their I'll try. I'll try to make sure my twos and disease look different. It gets a little complicated when they're right next to each other. Okay, So if X is my variable, I'm gonna have negative X squared, divided by two. And I have that Z already there in the denominator. Plus well, that becomes X squared over to just our power roll. And I'm going to evaluate this from X equals zero to X equals to see. Okay. So you can still see. I have one more integral to take care of, but let's get that Blue Integral resolved first. Okay, If I let X equal to Z or that first piece, I'll have to z squared on top of Tuesday on the bottom. So that's just negative to Z. Yeah. Okay. Plus, let X equal. Choosy in that second term, that's two Z squared over two, which is two z squared. Now, if I let x equal zero, both of those terms go to zero. This is all I have here. And to make life just a little bit easier, I'm gonna pull out, um, a negative to it's gonna pull out my constant through here, so that gives me Z minus Z squared. Oops. I'm sorry. I wrote DX. It's supposed to be D C. So there's my integral. Now I've got this one. I could just use the power rule integral of Z Z squared over two and Z squared is going to be Z Cube over three. And I would evaluate this from Z going from 1 to 2. Okay, so let's evaluate if Z is too. That's going to give me to minus eight thirds. That's that first piece, and I'm gonna subtract when I get if I let it equal one, that was gonna be one half minus one third. Okay, So in order to do this, let's just do a common denominator all the way across the board inside their common denominator of six. So I'll have 12 6th e have to multiply by to hear that 16 6th minus 86 This one's gonna be a plus 2/6. Okay, Just come up here. I've got just a little more room. So what do I have inside those parentheses? When when I do all of that math out, I end up with negative 5/6 inside there. And if I multiply that by a negative to that gives me a positive 10 6 or I can simplify toe a positive five thirds. So it's a lot of work. You've got to do this integration thing three times, but in the end, your answer will be five thirds.

All right, So we want to express this Integral in six different ways. Yes. We want to do this in six different plays. The first thing that we're going to do is going to say What if we just look at this hour are solid in the X y plain. So that means he is equal to zero. So we get this here. We've graft this here so you can see the bounds on X magnitude to two bounds on wise, You're in for minus X word. And then the bouncers of Z Well, it's just what happens, right? Bombs on Z here. Which happens when we solve for C here. Okay, we can see that we can put this into this into right integrating the Z. Then why then X? What if we switched these too? Hey, what if we do x that DX Dundee Y Well, we're gonna have to switch these two into girls. And we're going to look at this region not as region one, but as region, too. Okay, sir. That's what's going on there. And I'm this guy here is sold the same. So there's a first too. What if we look at it in the XXI plane. So why is equal to zero? So this guy's really just no lips as I've drawn here. So if we look at it as region one extra going from negative to two, why is going from sorry Z is going from ah, negative square this thing to posit Square to this thing and our why? Well, our wise just as it was given here. So let's write this Azan Integral. So since why is zero? So we go native to To to and we're going to do are ze bound from negative squared of one minus x squared over the four square root one minus x squared over four zero Tio, it's going to be four minus fact squared minus four z squared of f x y z So do I. And then, um and then we did a DX Correct? Yes, we did. A tx no Deasy and land E X. Sadly switch. These two integrations molars are switching our region one which went like this to region too in the region too, from negative one to one. I mean our bounds on X which are negative square root of four minus Z squared, tow for Mina's E squared square. Through that, uh, these bones are still the same. Effa x y z d y is still the same that we got d x d z He's got switched. All right, so that goes there. The foreign tools we need two more. All right, now we're looking at the y Z plane, so X is equal to zero. So that means wise equals zero and y is equal to four minus Z squared. Okay, this is our why, And this is R Z. Our graph. Looks like this here is for that's this guy. Okay, so, um, let's do it. Doesn't matter what region we do first, let's do it as a region One says you're in a four. And how are my eyes? He's going to be ranging. Um, well, we need to solve for C here, So let's see, Once we saw for Z, get for out. Sorry. Um, not that I get four minus y over four square eat and minus four, minus y over four. And then our expounds Well, it's when we solve for X in this formula here and so solving for X area four minus four z squared minus. Why, yes. And four minus four C squared Minus y. Okay, Now we need to put this into an inter girl. Negative one toe One for the Z grounds are y bounds like so? And then our expounds of X y z, we do do yaks. Then we do it. Why? And then we do it easy. So switching these two, we just have to switch from reading one to region too. So during that were going from zero to four around Square E. I'm sorry. I think I made a stake here. Yes, there should be. Give me one second here. So zero for Izzy bounder changing like this. Okay, that's right. On our expands or changing like this. So we ddx then we do. Why, yes, this is This is correct. Yes, this is correct. So then we want to change our bounds. Yes. We want to change our bounds to reflect the other one or minus y Over four or minus Wild for negative squared of four minus four Z squared monsters. Why squared of four minus four z squared minus y the vacs. Y z dx easy D y Yes. So this this here is is correct. Yes, this is This is totally fine. Yes. All right, we're done

The questions is that we have to express the integral triple integral over the region. Have of facts. Come awake mindset D. V as an integrated integral in six different forms. Where easy, solid bounded by a given surface By square plus, that is quite equal to nine. X equals 2 -2. This is nine and X equals to two. Now moving towards the solution, your first integral would be integration from minus 2 to 2. Integration from minus 3 20. Integration from minus route nine minus present Square to Route 9-. Is that the square export X comma, Y comma set dy desert D X. Now moving towards second integral, which will be integration from minus 2 to 2. Integration from minus 3 to 3 integration minus route nine minus y square to route nine minus y square D. Then dy dx here will be F X comma by comma. Is that third integral will be integration from minus 3 to 3. Integration from minus 2 to 2. Integration from minus route nine minus X squared to route nine minus six square f fourth. Ex coma like Marcel day by day, X day is it? You are fourth integral will be integration from minus 3 to 3. Integration from minus 2 to 2. Integration from minus route nine minus Y squared to nine minus y square half of X. Comma, Y common. Said they said the X Day. Why? Our fifth integral will be integration from minus 3 to 3. Integration from minus route nine minus X squared to route nine minus X squared. Integration from minus 2 to 2 F of X comma Y chromosome dx dy day is it? and the 6th. And the final integral will be uh integration from minus 3 to 3. Integration from minus route nine minus by square to route nine minus. Why. Square into integral from minus two to F of X. Comma Y chromosome dx DZ D. Why? Thank you.


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