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8 Find the volume of the tetrahedron bounded by the coordinate planes and the plane x +2y + 32 6_...

Question

8 Find the volume of the tetrahedron bounded by the coordinate planes and the plane x +2y + 32 6_

8 Find the volume of the tetrahedron bounded by the coordinate planes and the plane x +2y + 32 6_



Answers

Volumes Find the volume of the following solids. The tetrahedron bounded by the coordinate planes $(x=0, y=0$ and $z=0$ ) and the plane $z=8-2 x-4 y$

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.

Right. So you want to find the volume of this tetrahedron enclosed by this plane? So Ah, what I like to do is just imagine that my Z coordinate is Europe. So, uh, this is just the ex wife plane. And if zero then we have this line here. This line is two X plus y is equal to four. Okay, So, thie, why intercept appear is zero comma for and the X intercept down here is a two comma sirrah. Okay, so we're going to integrate this with respects to Why? First? So the integral is from zero to. That's the X values the lowest. Why value a zero and the highest Y value is this guy here, which is four minus two x ray. After solving for why and then I've got my sea right. My Z is equal to four minus two x minus Y four minus two x minus. Y remember, I'm doing d y and then d z Andi ex excuse. So then we've got zero two two four. Why? Minus two x y minus Y squared over two from zero toe, uh, four minus two x. So we've got four times four minus two X and then we got minus two X times four minus two x and then we've got minus. Let's see here we've got minus four minus two x All squared over two d x. And when we plug in zero right, the zero will just make everything. Here's Europe. Okay, so let's continue. So what we want to do here is just kind of simplify this stuff. So we've got sixteen minus eight X minus eight X plus four X squared minus one half times sixteen minus eight. Sorry, that's not going to be eight. That's going to be sixteen x plus four x squared at the works. So we've got sixteen minus sixteen X plus four x squared minus a plus eight ACS minus two x squared D x. So the integral from zero to two of eight minus eight packs minus started. Plus two x squared T X. Who out? The two four minus four acts plus X squared D X. So that too, in general from zero to two of X minus two squared d x. Okay, so it looks good so far and this ends up being X minus two all over three cubed. And this is equal to two thirds zero minus eight over history. Which ends up being, uh, the answer that we want. Excuse me. Sorry. Let me go back here and do this a little more carefully. So we've got two thirds times evaluating from zero to two. So this is going to give us negative eight. And it's going to this negative eight sounds, right? No. When we plug into, we're gonna get zero. When we plug in zero, we're going to get negative eight, so this ends up being sixteen over three.

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.

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.


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