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Evaluate JIf xzav where E is the solid tetrahedron with vertices (0,0,0), (0,1,0), (4,1,0), (0,1,1)...

Question

Evaluate JIf xzav where E is the solid tetrahedron with vertices (0,0,0), (0,1,0), (4,1,0), (0,1,1)

Evaluate JIf xzav where E is the solid tetrahedron with vertices (0,0,0), (0,1,0), (4,1,0), (0,1,1)



Answers

Evaluate the triple integral.

$ \iiint_T xz\ dV $, where $ T $ is the solid tetrahedron with vertices $ (0, 0, 0) $, $ (1, 0, 1) $, $ (0, 1, 1) $, and $ (0, 0, 1) $

The solution to the question 115 ways here We have to solve the integral integration from zero to to into integration from 0 to 2 minus x. And to integration from 0 to 2 minus x minus five Y square dessert day by day X. Now first solving with respect to that you will get integration from 0-2 into integration from 2 -1. Why square is that limit? Going from 0 to 2 minus x minus by day by day X solving this. You will get your answer as integration from 0 to 2 into integration from 0 to 2 minus x. Two Y squared minus x Y squared minus Y que day by day X. Now integrating with respect to y you will get your answer as integration from 0 to 21 minus three to minus X to the power four minus one by four to minus x to the par four D X. Now integrating this integral you will get one by 12- of 2 -1 to the power. five x 5 Limit from 0-2, putting the value, you will get your answer as eight x 15. Thank you.

The solution to the question 115 ways here We have to solve the integral integration from zero to to into integration from 0 to 2 minus x. And to integration from 0 to 2 minus x minus five Y square dessert day by day X. Now first solving with respect to that you will get integration from 0-2 into integration from 2 -1. Why square is that limit? Going from 0 to 2 minus x minus by day by day X solving this. You will get your answer as integration from 0 to 2 into integration from 0 to 2 minus x. Two Y squared minus x Y squared minus Y que day by day X. Now integrating with respect to y you will get your answer as integration from 0 to 21 minus three to minus X to the power four minus one by four to minus x to the par four D X. Now integrating this integral you will get one by 12- of 2 -1 to the power. five x 5 Limit from 0-2, putting the value, you will get your answer as eight x 15. Thank you.

Were given a triple triple integral and were asked to evaluate it. This is the triple integral over the region t of the Function X squared TV where t is the solid tetrahedron courtesies. 000100 010 and 001 First, let's sketch a graph of this tetrahedron so we have X Y and Z axes. I mean at the origin. And we have Vergis is at the origin courtesies at 100 Protect 01 zero never takes it 001 And the Vertex Well, that's all. Okay. And so we have it's triangular face here and then we have behind it. These three lines meet at the Vertex, and so this is our region t, which we're going to be integrating over. Let's describe this in terms of triples, we see the tea. This is thesis of ball triples X y Z, such that in the picture. Let's take X to be between zero and one. Then it follows that why is going to lie between zero and one minus X, and finally we see that Z has to lie between the planes zero and plane, which has which intercepts the X Y plane along the line X plus. Why equals one? So this is going to be the plane one minus X minus y. This can also be found by recognizing that this plane is the same as the plain X plus Why plus Z equals one, since it has X, y and Z intercepts of one. Therefore this triple integral to be evaluated as an iterated integral. So first will integrate with respect, dizzy and with respect to why then, with respect to X. So we have the integral of Mexico 01 integral from y equals 0 to 1 minus x integral from Z equals 0 to 1 minus x minus y of our function X squared d c d Y d x First we'll take the anti derivative with respect to Z and then substitute doing this in one step, we get integral from 01 integral from 0 to 1 minus x um X squared times one minus x minus Y do you? Why the x And here you'll simplify this to the integral from 01 integral from 0 to 1 minus x of X squared minus X cubed minus X squared. Why B Y D x and here we'll take the anti derivative with respect to why so we get integral from 01 of X squared. Why minus execute Why minus one half x squared y squared from 0 to 1 minus x DX substitution gives us the integral from zero toe one um X squared times one minus x minus X cubed times one minus X minus one half X squared times one minus X squared gxe. And to evaluate this, this simplifies first to the integral from 0 to 1. Um, multiplying things out. We get one half x to the fourth and we have minus X cubed and plus one half x squared. The rest of the terms cancel out DX taking the anti derivative. We get 1/10 extra 10th minus 1/4 next to the fourth plus 1/6 X cubed from 0 to 1, and substitution gives us 1/10 minus 1/4 plus 1/6 which simplifies to 1/60. And this is our answer

Okay, so for these let's p which is on arbitrary point. Let's define it to be x one x two x three So we need to find we need to find Barry centric What he needs off p With the respects to vatis ese given by you have a more supply by Iwan, which is 100 Be more supply by it. So which is 010 see multiplied by eat 3001 under then Vento zero, which is equivalent to scene, is 00 Zero b zero 00 c 000 No, we see that's p equal to x one x two x three I can ride is US X one over a more supplied by a 00 You're going to see it in the moment. Loss x two Overby off zero B zero plus X terry oversee 00 c plus. I can pick a constant key off 000 If you calculate this, you see that disappear. Call to X one extra Extra because it will cancel a Buick wants to be civil. Cancel. See on then, Kate and zero is still zero. So this implies we're K. I'm just right. It's were kid can be any number can be any number no boats since we need Okay, so be often combination off eight times You want beaters e to see Time's eatery on zero, then from lemon them this equation star Because actually, this is a Times B one beaters Eat through Citizens Eatery on and this is zero. So who wants the The coefficients? You want them to? Something equal one. So it will. Then we'll need that X one over a plus x two over B plus X Terry overseen plus que must be in court to warn. This implies that Kay is just move everything to the restaurant's x one over a minus. X two over B minus. X Terry oversee So that implies that the Barry Century warden needs, uh, excellent over a ext sue over b ex Terry Oversee on then r K, which is one minus x one off a minus. X two over B minus. X TERRY Oversee


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