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Find the indicated volumes by double integration.The first-octant volume under the plane $x+y+z-4=0$

Calculus 3

Chapter 29

Partial Derivatives and Double Integrals

Section 4

Double Integrals

Partial Derivatives

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Let's draw X Y plain. We've got the circle from negative to Teo too. But know that we're only looking at the first oxidant, right? The first talk tint Is this guy here? Because this is the first often so we see that our X is going to rain from zero to two. And our why is going to rain from zero to the square root of four minus x squared. That's just by solving this guy for why good and our double into girl. So let's get done. Black are double intro going from zero to two from zero to square root of four minus x squared. And, um, we've got the surface above as equals X y and below x minus R x plus y. So this is the surface that's above or on top and below like that. So what we're doing Do I tx So from zero to two, we've got X Y and excuse me, we don't Yes, we have X y plus why squared over two minus X y squared over two from zero to the square root of four minus x squared DX. Okay, uh, now we just plug in amusing FTC here plus one half of four minus X squared, minus tax over to for minus X squared, Dex. All right, so this is just a very standard and to go problem, this guy here we're gonna have to use use of. So when we do use of, we're going get zero. You get a negative out, we're going to get a four to zero to the square root of you. And you've also got a one half out here to you. Plus the integral from zero to two. Two minus x squared over to minus two X plus X cubed over too DX. Okay, So what is our next step? Well, we need to integrate this, right? So integrating this is not too difficult. We see that was that one half. We've got two thirds can four to the three. Oh, her too. Plus two x minus, X cubed over six minus X squared plus X to the fourth overrate from zero to two. And also done. We'LL get thirty two over three

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