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Evaluate the integral by changing to cylindrical coordinates.

$ \displaystyle \int_{-2}^2 \int_{-\sqrt{4 - y^2}}^{\sqrt{4 - y^2}} \int_{\sqrt{x^2 + y^2}}^2 xz\ dz dx dy $

0

Calculus 3

Chapter 15

Multiple Integrals

Section 7

Triple Integrals in Cylindrical Coordinates

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if want to figure out this integral using cylindrical coordinates, we'LL figure out how to change these bounds appropriately. So we have squared of X squared Plus why squared is less or equal to Z less than Abel. The two sets working with this guy here X squared plus y squared is R squared in ours Positive. This is the same as our is less than equal to Z is less than people too And or here we have minus square root of four minus y squared less than equal to x less than equal to squared of four minus Why squared So in political cornice this is minus squared of four minus R squared sine squared data less than or equal to R close signed data less wrinkled Do square root of four minus R squared sine squared data. Okay, so then if we square everything we get r squared co sign squared data is less or equal to four minus r squared sun squared data and then we can add r squared sine squared faded on both sides And then since sine squared plus co sign squared is one that would leave us with r squared is less than or equal to four and again ours, something that's positive. So this would this would give us that zero less than Abel to our is less or equal to now. We need to figure out what to do with data. So it looks like why is allowed to be minus two and it's a lot to be positive, too. So it seems like there's not really going to be much restrictions on Fada. Why, Khun b, you know all the way up. But it's Max of two. Always at the men of minus two could be positive or negative. The X values can also be positive. They can also be negative. So X and Y values can be positive or negative, so authentic and basically be anything. But we don't want to repeat ourselves. So we left. They'd a lie between zero and two pies so that all the angles happened exactly once case and that we have this set up. We can write everything so integral from zero to pie and go from zero to and a girl from our all the way up to and then we had X Z. So X is our cosign Fada and then Z is still Z, and we always want to multiply by our when we switched to cylindrical coordinates. So make that an R squared, and then this guy was the bounds we put on scene. This was the bands we put on our This is the bounds we put on data. So that's what we end up getting there. Okay, So integrating this with respect, who we get one half r squared co sign data is he squared and we evaluate z from are all the way up to two. Okay, so this turns out to be an integral from zero to high. And a girl from zero to two are squared, cosign, theta minus hard on the fourth over to cosign data. We are data. And what we might see here is that eventually we're going to have to be integrating co sign of data with the respect of data. All right, that'll happen eventually, as we have it set up right now, we would be integrating our first, but eventually we're gonna have to be integrating with the respect that data and looks like we're gonna be integrating a co sign when the ingrate co sign with respect. Oh, data. We're going to get in sign. And when we plug in zero or two pie and a sign, we get zero. So if we think about this a little bit, we might suspect that we could save a little bit of time if we just switch these guys and wrote were not a problem like this. So instead of taking the integral with respect our first, we could also take the girl with respect to data first and just, you know, switch these guys up. Okay, So integrating co sign of theta With respect to theta, we get sign and again when we plug in zero or two pie and for theta sign of that value is going to be zero. So this turns into zero teo zero p. R. So we just get zero.

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