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13 Use change of variables t0 find the volume of the solid region lying below the surface and above the plane region R: region bounded by the graphs of x -L xy-Ix-L...

Question

13 Use change of variables t0 find the volume of the solid region lying below the surface and above the plane region R: region bounded by the graphs of x -L xy-Ix-Lx-2 (Hint: Let *="y=vl" ) Round your answer (0 tWo decima places. Important t0 sketch the region of integration . belore and alter the transformation (10 points)

13 Use change of variables t0 find the volume of the solid region lying below the surface and above the plane region R: region bounded by the graphs of x -L xy-Ix-Lx-2 (Hint: Let *="y=vl" ) Round your answer (0 tWo decima places. Important t0 sketch the region of integration . belore and alter the transformation (10 points)



Answers

Volume of solid
the volume of the region in the first octant enclosed by the cylinder $x^{2}+z^{2}=4$ and the plane $y=3 .$ Evaluate one of the
integrals.

Divergence of F. It's equal to three white squared blood. So plus 30 square. So using cylindrical Cody needs in cylindrical to coordinates. All the needs. We have Y. two. We go to our course. See it sir. Then Z. To be equal to I was saying teeter then eggs to be called to ace. So then this implies that the double interior over us. F. D. S. It's going to be the triple in Segre over E. You have three wise squared plus three zero squared D. V. And this is going to be cool. So DNC girl from 0 to 2 pi There are 2 1. Then from -1 to to you have three R. Squared course square teeter Plus three hours grade sign square seats. Uh You have the R. Are the uh the so we have ah the ex the are the seats. So this is equal suit three is common. So three as a constant out from 0 to 2 by the pizza. Then from 0 to 1 for our Q. It's the R. Then from negative 1-2. You have the X. Because cause square teacher plus some square teacher, It's going to give us one. So then this is equal to three to buy. This will give us one divided by four same story and a final and science equal to nine by divided by so.

So we have our function and we have our region, which is going to be a cylinder to find by the circle. Why squared Plus C squared is equal to one and X is equal to negative one 10 x is equal to negative deposit to the first thing we're gonna do is find her divergence. I m f just so we can convert our flocks formula into a volume integral instead of finding the flux through all of our services. So our divergence of f is gonna be d over. Yes, times are acts which after the three x y squared plus de over d y tax e esteem over D. C uh, Z cute. No, Our first partial derivative is just gonna be three twice where their second isn't zero and then our third is going to be okay. Three z square or we just see this will be equal to three y squared, plus the square. No, because we have a cylinder. We're not gonna be able to find our volume from just our Cartesian coordinates. We're gonna have to split it up into cylindrical coordinates. It's a political coordinates normally used Z, as are variable to stays the same, but in this case, we're gonna want extra Stay the same because where our cylinder is going down the X axis so well defined or why, as our co sent if I and rz as our sign If I We use this because we want a circle rotating around the X axis which we have our region to find as and then we'll call her differential volume is gonna be our If I they are then d x this time, Not easy. So now we just convert part of virgins into cylindrical coordinates. So we have three. That's why Square, what's the spur Just equal to three times our co sign If I squared plus our sign If I squared And now if we factor in an r squared we're gonna be left with co sign square to five plus sine squared If I which is just equal toe one these are divergence It's gonna be equal three R squared No, we can calculate our flux. So our flux equal to the surface interval of f times, Yes, but a divergence theorem that's equal to the triple integral of our volume on the diversions of so now we have everything to find a way to do the bounds so we don't get ahead of ourselves. And then we have the divergence, which is just three r squared. They're differential volume, which is our defy Do you are easy The axe in this case. So we have no reliance on fi or ex inside are integral so we can just factor was out It's on the X integral, which we have to find from our region is negative one. It's a positive too ar fi which is just seared to pie. It goes all the way around the circle and then our our for our bounds can be defined by this circle we have here. That's why squared plus c squared equals one. So we know that this is a circle equal to the radius squared. So we have one is equal to r squared, so our radius is gonna go out to one now, inside we have three times are cute d r So we just take our first and it roll. We have two minus inevitable. Second in the role she's gonna be two pi. My zero third in a rule is just gonna be three are to the fourth divided by four, all from 0 to 1. So we have three times two pi times three 3/4 of one to the fourth by zero fourth for just 3/4. So now we multiply all these together and we get three times three In the numerator. She's nine. The two cancels out with the four to make that two, and then the pie is multiplied on top. So we're left with nine pi house.


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