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Compute the volume of the solid bounded by the cylinders z = y , % = yP and the planes z = 1, =16/532/152/316/152/5...

Question

Compute the volume of the solid bounded by the cylinders z = y , % = yP and the planes z = 1, =16/532/152/316/152/5

Compute the volume of the solid bounded by the cylinders z = y , % = yP and the planes z = 1, =1 6/5 32/15 2/3 16/15 2/5



Answers

Find the volume of the solid in the first octant bounded by the cylinder $ z = 16 - x^2 $ and the plane $ y = 5 $.

All right. So we want to go ahead and calculate the value of this solid. Ah, founded by our by selling their the equal of 16 minus X squared. Ah, and the plane. Why? It was five in the first octane. So since we're only working in the first docked in, we want to find our our other bounds. So we know, um, because of our restrictions, we know a couple of balance should be the origin and X Y, um 05 which is the the plane y goes five intersecting with the with the X axis and the Y axis. And, um so we want to find our ex bounds. So we know that, um, this Ah, this cylinder in the first octane should be bounded by, um, Z equals zero. So then we find a solution. When Z equals zero, we get that Ah, X squared is equal 16. And in the first act in this only applies to X eagles for so then we know where other points should be. 40 and 45 or our region are is our rectangle 0 to 4 and 0 to 5. So then we can go ahead and find the volume so the value is rest of l one to grow over our of the d A or, uh, the integral or our of 16 minus x squared D A. Since we only have ah, x is we're just gonna go ahead and a great respect to X first. So then we have our outside bounds and her inside bounds of 16 minus x squared the axe. Why? And this is pretty simple. Evaluate. We have keep the outside the same. None on the inside we have 16 x my A sex Cuba for three from 04 And since both of these air zero at zero, we just have Did you go from 0 to 5 of 64 minus or cured over three, which is 64/3. The, um, you. Why? So then now we just left What we're left with This girl is 0 to 5 of 1 28/3 Do I, which just gives us Ah 1 28 times 5/3 or 640 divided by three. So this is the total volume underneath our cylinder

Looking at the first Ogden's. So let's find our bound. So when X is equal to zero well, we've got why is equal to plus or minus one. Okay. And, um when why is equal to zero? We've got X is equal tow plus reminds just by given this these guys here. So we want our double integral to see that our plane Z is equal. Teo, Why? And we are either going to integrate with respect to X or wife. So I'm going to choose integrating with respects to Why first, So Mei Wai bounds Well, the lowest my Why, Khun b a zero, right? Because we're dealing with the first often and the same thing with X so x. I'm going to say it's gonna vary through one and a zero in one. Right? That's the max value of one. The right most value of access is one. And for why? Well, that's going to depend, right? It's going to depend on X. So what I did here is so we're a lot. So I've got why is between square root of one minus X squared and zero and exes between zero and one. All right, So, uh, going ahead and integrating this zero to one y squared over two from zero squared of X one minus x squared D x. So what we end up with is one half times one minus X squared the ex pull up the one half and we integrate from zero to one. Get one half one minus one. Uh, I'm sorry. This should be one third. Yeah, and, um, Let's see here. I did this directly. So, yes, this is two thirds. So one half time's two thirds is just one thing.

So in this question, asked to find a region, but it by the given care. But so go ahead to town. Uh, ex guy. Plan reaching. Thanks. Why? We have exits, but by one. And juice. So ones. Yeah. Juice. Yeah. So have it's good. Yeah, just another lie. Yeah. And now we have another Jew. Girls. Why could you one of the acts? So one of the Xbox mine desk And why would you? Minus one over x o Jeff livid over another one? Yeah. Yeah. So the original we're looking for in the ex Quite plan will be in this question. Yeah, yeah, yeah. Okay, so and we have the why. Hey, could you one Thanks. Yeah, we have the one you could you minus one of the X on Ex government wanted to. So we're ready to formally about the core now. So the inside we have the the upper ah cover will be express one man Isdell, our company's Oh, so which express one? And now yeah, we had to see which would be easier for us to send the order now, so we must easier to do the do I first to do. Why the eggs. So do you write One Goes formed a lower cuff A book of So we'll be of Rome minus one about thanks Ju one over X on X A girl from your memory will be one Don't get him X women Could you do So we have is an equal a chew from one to draw again So now why is a constant what I wise a verbal So exit constant So have expressed one And now they're limited Inside we have one over X minus, minus one over x x So heavy isn't going to do someone to do express one The number of terms will be too Who can put you outside on now? Over X Thanks present in Cochin Jew, I want you to do so. We have one blessed one other. Thanks. Okay, thanks. That isn't restrained from which is going to hams. The groom wondered what? Your ex in the girl? No, one of execution and land off absolute thanks. And now the commitment from wanted you. So it could you do so if we would join saying we're on a job close on a you know, minus Don't know remit one when he said would be one minus one. So why don't you? Could you do inside? Can do minus 1 to 1 plus on a two and Elena one pitches room. So we got you. Could you? You plus two on a Jew and that's a fine I understand.

So in. So in this video, we are told to, uh you cylindrical coordinates in order to find the volume of our solid which is in the first Occident. And it's bounded by the cylinder of radius one. So our radius is one. And also it is bounded by the by Zizek auto accident scenes. So the first thing that's analyzed with this first off millions mean so first talk, it means that our X and Y values have to both peoples Well, in what or what angle will force our X and by toe always be positive. That's our angle. That's from zero to pi over two. So this first often helps us determine data. So data has to be between zero and high over. Okay. Z is given to a Z has two is equal to zero So rz is bounded from zero to x. But remember, we are in cylindrical coordinates are exes are co sign and are ours also easy. We're told that the radius is one that means our varies from so again rz goes from zero to off. Sorry. So get rid of this. We know that X So this goes away. So it's from zero to our coastline data our our it's from 01 on our data because we said first often are X and y has to be positive. It's from zero to pile. All right, now we know that volume is simply there triple into growth TV and DVR volume element is our disease, Yardy data, because this is so powerful Corvettes, and now we have to write our limits of integration correctly. So the first thing we have is D. C. We know, see varies from zero to arcosanti. So the innermost integral its limits are zero in our coast. Next we have your so then are are so their second into grow Berries. But you're a one. And then finally we have state us. Our outermost integral has to vary from zero to all. Right now that's that's compute this interpret the intervals are with respect. Izzie is our time C where c where, uh, Where z goes from zero to our coast. All right. Now we plug that in, so it's just far times Arco Science data zero, which is just R squared. Okay, Now we have to take the integral off R squared coastline data with respect to our so coastline data is just a constant with respect. You are so really we only have to take the integral off our square. So that's just co science data times are pew divided by three at our limit of our limits are from 0 to 1. So that's just coastline data times 1/3 so we can pull that 1/3 to the outside. Now we need to determine the integral for science. Data on the limits of integration are from zero to pi over two. The integral of coastline Beta signed Fada Data goes from zero to pi over two. Now we plug that in its 1/3 times. That was signed by over tools one sign 00 So it's 1/3 times what might zero, which is one thing. So the volume of our solid is 1/3


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