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Problem 6_ Find the hypervolume under the hypersurface w = xy over the solid that lies below the plane 2 = x + y and above the region in the ry-plane bounded by the...

Question

Problem 6_ Find the hypervolume under the hypersurface w = xy over the solid that lies below the plane 2 = x + y and above the region in the ry-plane bounded by the parabolas x y2 and y = 22 3/28

Problem 6_ Find the hypervolume under the hypersurface w = xy over the solid that lies below the plane 2 = x + y and above the region in the ry-plane bounded by the parabolas x y2 and y = 22 3/28



Answers

$23-32$ Find the volume of the given solid.
\begin{equation}
\begin{array}{l}{\text { Under the plane } 3 x+2 y-z=0 \text { and above the region }} \\ {\text { enclosed by the parabolas } y=x^{2} \text { and } x=y^{2}}\end{array}
\end{equation}

Were given if a vector field F in an oriented surface s were asked to evaluate the flux of F across S so f is the vector field X I plus yj plus five k and s is the boundary of the region enclosed by the cylinder X squared plus Z squared equals one and the planes y equals zero and x plus y equals two. So if you can visualize this region, we noticed that our surface s actually consists of three distinct surfaces. So we have s one which is the lateral surface of the cylinder. And then we have s to which is the front, so to speak of the cylinder formed by the plane X plus y equals two. This is a vertical plane and we have s three, which is the back, so to speak. Formed by the plane y equals zero another vertical plane Now on s one. Our vector field f in terms of the surface which we parameter rise using data And why, since it is a cylinder Well, this is going to be I want to premature eyes The cylinder using X equals Kurtz Cosine theta. Why equals why and z equals sign of data since their cylinder has a radius of one and therefore we have that are vector field. We have X I they're actually a few different ways to do this. What's premature? Eyes it with X equal Sign data and then Z equals cosine theta instead. Then our vector field has X I which is thesis ein of Fada. I plus why j which in this case is just why j still and plus five k so Z doesn't play a role at all. And from our cylinder using the equations for a cylinder, we have that the cross product are thate across our Why this is going to be sign data I plus cosine data. Okay, notice that this is actually the same as the position vector Except for now, why is equal to zero? This is a property of cylinders. Then we have the flux of f across s one. This is the iterated integral and we're using our parameters. Ation here from faded equals 0 to 2 pi and it is integral from why equals with a plain white equals zero to the plane y equals tu minus X. But we have the X is signed data with this premature ization. So to minus sine theta of a vector field started with the normal vector the outward normal vector that is. And so we get sine squared data plus and then zero times Why? Plus five times cosine data D Y d theta and taking the anti derivative with respect to why this becomes integral from 0 to 2 pi of to sign square data plus 10 co sign data Yeah, minus sign Cube data minus five. Signed data cosign data together terms or zero d theta. I'm taking the anti derivative. With respect to theta, we get rather complicated expression which is not impossible to find. I'm not going to write out here. Mhm. We end up getting two pi. Yeah, Now looking at surface s to this is the front plane. Well, here we have that our vector field f is going to be evaluated on the surface which we parameter rise with x and Z. So this is going to be while the surfaces parameter rised as ex I plus And then why is equal to minus X j plus zk? And so our vector field becomes X. I still plus y J where why is now to minus X J plus five k So Z doesn't play a role again And we had that the unit normal vector rz across our ex facing outward from our region, this is going to be simply the vector I plus J. It's actually not the unit Normal vector. Let's a normal vector that's facing outwards And so the flux of f across s to this is the double integral over the region X squared plus C squared is less than equal to one. So the projection onto the XY plane of F dotted with our normal vector. So we get X times one plus two minus X time is one plus five times zero d A. And this simplifies to double integral over this disk of to in this disc has an area of pie. So this integral is simply equal to two pi by geometry. Finally on the surface as three he back face we have that this service could be premature ized as simply are of X Z equals zero I plus zero j x i plus zero j plus zk Since the plane why equals zero and so our vector field is F f r of X z, this is equal to X. I just still x I plus why j But why is now zero? So it's plus zero j plus five k. So once again, Z does not play a role. Yeah, and we have that the unit normal vector or just a outward normal vector I mean, from the surface is going to be facing in the negative y direction. So negative J would be an example and therefore it follows that the flux of F Across S three This is going to be double integral over the disc X squared plus C squared less than a record one of f dotted with the normal vector which we see from the equation. This is the double integral of zero, which is simply going to be zero. This contributes nothing to the flux and therefore the total flux of F across s. It's going to be some of these individual flexes, which is four pi

First of all this sketch this area and second organize fund this Bolland by integrating it with respect for s and why so, axis ranging from literature to one here and why's ranging from likes to eggs and likes to two minus x squared, which is a problem here. Yeah, basically this in this over region right inside it in this part is way Who s and as faras Wake was too two months and swear. Two months and sweater sort of all Americans too Next to you One extra, too much explored. That's squared E Y yes, interval. Why we have next to one. That's why thanks to those two months and squirt years and after calculation this sexually two or three ex cute miners, that's the five over five plus servants miners. That's the full for from next to one, which is, he was too 63 or it's funny

Here in this trouble and we have to find the volume of the solid, which is formed by the slender Z Z equals two X Square on the parable y equals two to minus X square on the line. Why Quills? Two x So the volume would be quilts to the double Androgel are X Square D A. Where r is the region bounded by the Gulf's Why cools to do X Square two minus X squared and why quilts to X. Now I'll be loving these to go on a graph paper to write the limit for the integration. So let's telegraph able now and I'll be sketching this golf So there's no access on this is X X is and this go over represents the battle below, which is why gold two minus X square on this land, the present equation y equals two x on the reason of indignation would be decision. Now I'll be riding the limits for that. I'm taking award Eagle Strip so we can write this integral as a double integral X squared D expert, because to do adx onda limits for divide would be from here to help. So here why calls to X on here. Why equals two to minus X squared and d expert Very from this lane to this see equation of this lane is a X equals to one on The question of this line is X equals too negative, too. So fine. African right. The limits for ex ritual big worlds too, too negative, too. 21 I'll be simplifying the inner and wriggle first. So negative 2 to 1. Harry, get extra square. Why? And the limits are from X two to minus X squared daughter BX Assembly I will planning the limits. So negative 2 to 1 hair brigade X square times two minus X squared minus X cube Notter DX This candidate and now is the integral negative 2 to 1 hair irrigated, two X squared minus its extra power four minus X cube. Dr DX Assume V I. B. Integrating this integral. So we'll get negative to sorry. Here there's a bracket there on it will become two x cubed, divided by three minus X to the power five divided by faith minus x It about four divide before and literature from negative to negative 2 to 1. Not simply I'll be going to plug in a limits, so I get the one. But sorry. Black says. One will get to buy three minus one by five, minus one before minus when black value as negative too, will get negative to cube times. Chu is negative. 16. Divide by three He ever get started to buy Fife minus 16. Divide by four after a simplification will get the result as a 63 divided by 20. So the reason off. So the William that his office configuration ID equals 2 63 by 20 So this is the final answer.

So if it's probably want to find the volume of a good and solid eso, we want to take a double. Integral. Um, we know that the X value in this case is going to be from 0 to 1 in the UAE. Ultimately, what will be easiest first will be to solve, um, the equation for Z. So we do that we end up getting the Z equals three x plus two y on Ben. We know that's bounded, so we have zero toe one and then we have the integral from X squared to one half because we know from 0 to 1 extra, one half is actually greater than X squared. And then this is going to be our Z value in here. So have three X plus two y and that's a B B y. The x, um so the first thing we can do, he's just differentiate this inner portion on. But we get as a result of that is that this is going to be all equal to three x to the three halves minus x to the fourth minus three X cube plus X on data will ultimately give us 0.75 And the reason why is because we can differentiate this. I'm using anti derivatives and we'll end up getting six halves and it's or 6/5 minus 1/5 minus 3/4. That's one half 0.75 Or if we simplified the fraction, we would end up getting 3/4 so that would be our final answer in the volume of the solid. So because its volume, this will be Units Cube.


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