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Evaluate the triple integrals over the indicated bounded region $E .$$$iiint_{E} y d V, ext { where } E=left{(x, y, z) mid-1 leq x leq 1,-sqrt{1-x^{2}} leq y leq sq...

Question

Evaluate the triple integrals over the indicated bounded region $E .$$$iiint_{E} y d V, ext { where } E=left{(x, y, z) mid-1 leq x leq 1,-sqrt{1-x^{2}} leq y leq sqrt{1-x^{2}}, 0 leq z leq 1-x^{2}-y^{2}ight}$$

Evaluate the triple integrals over the indicated bounded region $E .$ $$ iiint_{E} y d V, ext { where } E=left{(x, y, z) mid-1 leq x leq 1,-sqrt{1-x^{2}} leq y leq sqrt{1-x^{2}}, 0 leq z leq 1-x^{2}-y^{2} ight} $$



Answers

Evaluate the triple integral.

$ \displaystyle \iiint_E e^{\frac{z}{y}}\ dV $, where
$ E = \{(x, y, z) \mid 0 \le y \le 1, y \le x \le 1, 0 \le z \le xy \} $

You don't evaluate this triple into go, and we're going to use the fact that do you over you squared plus a squared people too. One of Ray Ark tan You over, eh? Let's see. All right. So setting up this triple into girl here just like so first downs or X one z, then why, All right, So, um s o integrating with respects to X, we're going to get Z Time's gonna make use of this formula here. It's going to be one over easy Orc can of X oversea from zero to see. Easy d y. Okay, so these guys were going to cancel it up just great. So then we're going to go for one, two, four. Um, why two, four? We're gonna plug in ze for X and we're gonna plug in your So then we still got a d z Do I? My arcane azure azure Zira in our ten of one. And that being pi over four, the pi over four into go. I want to for him to grow. Why before? Ah, of easy. Do you want these integration become really just a trivial for us? In fact, four minus Why? Um do I And then we'LL get for why minus y squared over two from one to four. Uh, evaluating this guy here, we won't get the following. So this is just a very, very straightforward evaluation, okay? And ah, not gonna, uh, bore you with simplifying this stuff here. Just get right to the answer.

So we have to first set up our our triple injury. Can sar expound You're going from zero to three. Are y Ballenger learning from zero tax and r z bounds. We're going from negative X minus Y x plus y t y. So again I'm doing d z do you? Why tx So first doing with respects to Z we're just going to get why times see as are integral that the wise the ex Okay, so after we put these Belgian, we're going to get Why Times X plus y Linus. Why? Times x minus y do you live? D x so nice Cancellation happens. Well, I get Why squared d y, Jax? Okay. Integrating that we're going to get Why cubed I looted from zero X uh, D x. That one third is up here now. Two thirds of to go from zero two, three of just Sorry, which me of ext X cubed Pontiacs. So you get two thirds times one fourth x two four zero three Uh, we'LL end up with one six times three to the fourth. It's one six times eighty one and we can let's see if we can divide this stuff here. So if we divide the top and bottom by three. Twenty seven over too

Okay. We want to integrate six xy over some volume and the volume is described over here and because it tells us this and this we don't really have to to draw the volume because it's saying go from Z equals zero, which is the xy plane to this? See okay, so that's what's going to be on the inside here. 0 to 1 plus X plus Y six x y DZ So all we really have to do is look in the xy plane and see what we're looking at. Okay, Y equals the square root of X Goes through 001142. Okay. So it looks like a parable on its side. Why call zero? That's the X axis And then x equals one. All right. So I think I would do Dy first and I would go zero to the square root of x. Okay. And then X will go from here. Zero piled them up until they get to 101 dx. All right, let's integrate that. It's 0-1 0 to the Square Root of X. There's no Z here. So six X Y is a constant. So we just have to stick a Z to it. X plus Y. Most squared X. Six X Y times one plus X plus y minus zero. Okay. I think you're gonna have to multiply the 6xY in there before you can integrate. So let's do that. Six X Y times one plus six X squared Y plus six X Y squared. Okay. Now we're integrating with respect to Y. So the first one will be six X Y squared over two. So three X Y squared. And the next one will be six X squared Y squared over two. So three X squared Y squared. And the last one will be six X Y cubed over three X Y. Cute. Okay. 02 square root of X. So 013 X squared of X squared. That's X plus three X squared squared of X squared. That's X plus two X squared of X squared of X to the third. Power upon a call that X to the three house zero. Okay, this one is three X squared. So it's integral three X cubed over three. This one is three X cube. So it's integral three X. To the 4/4. This 12 X to the one X to the three halves. That's two X. To the two halves, X. To the three have so X. To the five halves. So it's integral. two x 2. The seven house over 7/2. Yeah. Okay so we plug in the one and we get one plus three fours plus 4/7. When you move that around -0 0. Okay. Okay. I just got a common denominator of 28. So one is 28/28. This one is 21/28. This one is 16/28. Okay. 28 48, 59, 65. Over 28 is what I get for the answer.

Okay, so we're giving the triple integral of why Ellen X plus z. Okay, so we can see that with our bounds. That why is limited by our function of X. So we're gonna need Thio integrate with respect to y before integrating with respect to X. Okay, but let's start out with integrating with respect, Susie. So that's from zero 21 easy. And in the UAE and in the X. All right, So we have. Do you? Why's from zero to Ellen of X, and then the X is from Joe or actually want to eat? Okay, so it's integrate with respect to Z. What do we get? Get the integral from one e and a go from zero to a lot of ex and why? Oh, and, uh, ex I'm see evaluated at one end. Plus the squared over two. Evaluated at one and don't see why the ex worry off, huh? This is this. Why Teller of X times one minus. They're all citizens. Is why one of X plus 1/2 d y the ex. Now let's integrate with Inspector Y. You get the integral um, once e of y squared over to a little X evaluated at villain of X and Joe I'm don't have X plus when half why. I've awaited a Ellen of X. And so the X Okay, so this gives may Oh, and cubed of X over too. Plus Pullen of six over. It's you okay. And now it's a value This from one to E. It's pathetic. Expected T X. Let's start out by pulling out our 1/2 You have only three X plus Volatile ex, The ex head. What in a girl of Owen Cube of X is Exelon, cubed of X minus three X Ellen Squared of X was six x ellen of X minus six x. We've all waited a e and one plus in a girl of a lot of ex That's just excellent of X minus X evaluated at B and one. If you plug these values in and evaluate this, you get 1/2 you're going to e seven, which is equal to seven over to my necessity


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