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The volume of the tetrahedron bounded by the coordinate planes and the plane > # % # /: :4 is...

Question

The volume of the tetrahedron bounded by the coordinate planes and the plane > # % # /: :4 is

The volume of the tetrahedron bounded by the coordinate planes and the plane > # % # /: :4 is



Answers

Find the volume of the given solid.

The tetrahedron enclosed by the coordinate planes and the plane $ 2x + y + z = 4 $

In the question, we have to use the triple integral to find the volume of a given solid, the tetrahedron and closed by the coordinate plane And the plane. Two weeks plus y plus that is close to four. Now, moving towards the solution here, why is 000, two weeks plus zero plus zero will be equal to four. That sp so the points are too common. Zero comma zero. Similarly your points for Q will be zero comma for comma zero and four are will be zero comma zero, comma four. No the it can be written as two weeks plus Y plus zero is close to four. Why will be 4 -2 weeks? So integrating from 0 to 2 and integrating from 0 to 4 -2 weeks day by the X. So now moving further That is equal to 4 -2 weeks -Y. So integrating from 0 to 2 into integrating from 0 to 4 minus two weeks into integrating from 0 to 4 minus two weeks minus Y D said d by d X. Now we will be solving this integral and solving or integrating first with respect to X. You will get integration from 0 to 2 and two. Integration from 0 to 4 minus two weeks. Four minus two weeks minus why the why the X. Now integrating with respect to why you will get Integration from 0 to 2. Four into four minus two weeks minus two weeks into four minus two weeks minus one by two into four minus two weeks. Full square deeks. Now moving my back integrating with respect to works. You will get uh sorry solving the previous equation, you will get integration from 0 to 22 X squared minus a tax plus eight dx. Now integrating this you will get two by three x q minus four x square plus eight X. And limit going from 0 to 2 which will be equal to 16 by three. And so this will be our answer. Thank you.

Eight minus two eggs Minus Who? Why? Because too little A big plus two y it was, too. Store X people still four minus to boy boy Kate words toe minus one divided 12 Integral begins from 0 to 4. Integral begins from a little to to minus bond, divided by true geeks. Eight. Loneliness, 12 weeks at last More boy do you worry beaks equals two. Integral begins from Ciro Toe AIDS Integral begins from 02 to minus one. Divided by true X Dear Boy, minus in two weeks old. Begins from 0 to 2 minus one, divided by two x even really minus four in turbulent begins from zero to two minus one divided by two eggs. Roy de Voix Do we thanks? So it means the integral begins from Is there a little full eight? Why Betweens little aunt to minus one Divided by two x minus two peaks. Why beat beans? Little two marinas won't be avoided by two X to my Penis. No one wanted to, um minus world. Why? To the voice that boy to with being was little on, uh, two minus one divided by two ace the X because so integral begins from 0 to 4 81 thing led to minus one, divided by two weeks minus Tool Case Who seeks two Minus bon. Divided by two x minus two months with two minus bond. Avoided by WHO? X thes. It's, of course to integral. Begins from zero toe full. 16 a minus eight eggs, No sir, X two Minus aid. Four weeks Nos. One day. Going to play, too ace, too. The mission waas too. Eight digs. It means little arms full minus four x two. Divided by who it means little earns. More knows won't devoid of y two x three Divided wants tree. It means little terms before which becomes 32 the voice says Wait three.

All right, so we're gonna evaluate this triple integral to find the volume. So what I've done is we obviously know that since it's enclosed by the coordinate planes are Z is going to be bounded below by zero here and above by well, when we saw for Z here, next thing we want to do is we want to find the X and y bounds. Which are these the way we do That was we put it into a a a coordinate plain where z is era a k x y plain. We can see that our wise going to very like this because one zero zero write you get this equation to see that our Y bounds air like this and our expounds air like this. Okay, next thing we do is we set up this integral here, right? We got our expounds r y bound and r z bounds. Okay, X y z Well, you know, when you start to integrate integrating with respects to see, Since this is a one here, we just get this as our into grand. Uh then we integrate with respects toe why? And ah, you get this, Uh, get this right here, do some cleaning up some more cleaning up. Then this is just a cow. Quentin to go and we integrate and we get sixteen over three.

Okay, so they want us to find the volume of the tetrahedron um above that is formed by the coordinate planes and the plane given by the equation that can be written as Z is 4 -2 X -Y. Okay. And so what that looks like then as it lives in the first oct end here. All right. And so that that will will hit these points Here on the z axis at four. Over here, the y axis again also at four. And on the X axis over here it's the point to And so the part of the plane in the first Aachen is this triangular region here. And so behind that down to the origin there is et cetera federal shape. Right now this is the domain that we're integrating over here in the xy plane. And you can see here that corresponds to this plane when Z is equal to zero. So for example, that line um can be written say is why is equal to four two X. Right? That's that line right there. So to find that volume then we integrate from zero up to that plain up to this value of Z. Um and again, you know, so you could have done it either way but set it up here safe for X on the outer and a girl. So say that X will go from 0 to 2 and then fruits value of X. Um you know why will then go from my pad frozen and there we are From 0 to 4 -2 x. Okay. And then we integrate um Over 4 -2 x here like this minus why And in a great first over dy and then over dx. Okay. So then just carrying on then we got exco and 0-2 and then integrating this Um we have 4 -2 x. Times why minus um Y squared over two from 4 0. Mm hmm. Um That to both of those and then ultimately integrate that over X. All right. And so This is equal to um x equals 0- two. Okay, so before -2, X squared minus. Yeah. Okay. 4 -2 x squared over to crime. So It just all comes down to 1/2 Integral from Mexico all zero uh to to of 4 -2 x squared dx. Right. And so of course we can multiply that out if we want to and do it. But you know, maybe you save a little bit of time by doing a substitution. So let's go ahead and look at that. Notice that I can now I could say substitute you is 4 -2 x. So that D you would be -2 the X. And then this integral will become um all in all there were going to have one, say minus 1/4 From this -2 here, taking it over there. And now this integral will become and integral Over you. That will now go from four 20 plugging in the expression zero and 2. Um for um for X. There. Right? And then this is going to integrate U squared. Do you write? And so this comes out to be 1/4 Um U cubed over three From 4- zero. I have to change that stuff around. And ultimately then the answer is 16 3rd.


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