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The volume below the surface f(x,v) =x? sin v and above the rectangle defined by [0,3]x [O,n] equals...

Question

The volume below the surface f(x,v) =x? sin v and above the rectangle defined by [0,3]x [O,n] equals

The volume below the surface f(x,v) =x? sin v and above the rectangle defined by [0,3]x [O,n] equals



Answers

Use a CAS to show that the volume $V$ under the surface $z=x y^{3} \sin x y$ over the rectangle shown in the accompanying figure is $V=3 / \pi$.

Okay, so we're given the surface z equals X squared. And basically, we want to find the volume between this surface on the Z equals zero plane. So what I mean by that is we draw out the X Y c cordant axes. We're trying to find this volume between the Z equals zero plane and this surface. Okay. And we're looking for that of the volume between this surface and this surface for the region described by these sets of points. So it's the seven points X y such that, um, X is between two on wise between zero and five. So this little set notation and basically kind of reads like English, if you know what What is going on? Um basic. Oh, wait. Um, you rewrite this little bit. Think I messed stuff between? Sorry. Give me one moment. Okay. Let me rewrite this. Actually got this s so bad. Okay, One sec. So, basically, our region are is described by the set of all points, which are element of r squared. Such that zero x is between zero and two on. Dwight is between zero and five. So reading this in english, it's basically are, is the set of all points X and Y that air in R squared R squared is basically like the set of all set of all co ordinate pairs such that X is between zero and two and wise between certain five. So that's what our region, ours and our region are kind of looks like a tangles. So you have hi be this rectangular region here that we want to find the volume between this surface on the surface. So in order to do that, we set up a double integral, um, over function X squared. Ta on. And I'm gonna do with u x t y Integral because I just want to get the X out of the way first and now, nor to write the limits correctly. I just think about how each variable is ranging. So since we're enduring respect to experts on the inside, I think about how does X range? Because the region is stated in this way. We have X rating from 0 to 2 and that why ranging from 0 to 5. So this would be our double in April here and just so valuing this, it's very easy to of X squared T X. This is equal to X 1/3 over three. Evaluate from 0 to 2. She ate thirds my zero. And then we want to put this into the outside integral. So 05 of 1/3 t Why and then this should be. There's cereal five t y. And a little trick here is first not very useful here, but I like to do it. But basically, if you're taking the integral of the function, one of any variable doesn't matter if it's d Y d x DZ de road. Do you think that if I whatever variable you want shoes? Basically, this is always gonna be the length of this. So what I mean by that is this new rule is just basically the length of our interval. So it should just be 1/3 times five and usually noodles give you area. But in this case, it gives you the length of the interval and you'll learn later that the double integral of the function one D a over a region are is basically just the area of region are so whenever we're in agreeance a function one it the dimension of our answer always scales down by one. So if we're finding the area right and we're using a single integral, usually you would find the area. But if we're integrating the function one, it basically scales down one dimension. And now we're finding the length of the line and then usually for a double, integral, you're finding the volume. But if you're integrating the function one, you should just give you one dimension down or basically the area of our integration. So 1/3 times five is 40/3 on that is our final answer.

The program today all starts to find the very, you know, off distraction so different you move. This function is good too for you, for your rights for Yamazaki is go to imitate a girl that's good grades first From why? Because it is easier 0 to 3 inte grew zero 22 for Biggs squared across seven then the Biggs the way. This is good too. And they grow roses from 0 to 3. Integrating dysfunction off respect too X can have a single full X created press table off seven, which is goingto this is it. Full X cube over three. Remember that the technical fixes creates go to x ray A press to press were over to persuade six cubic over three press seven x poses a question Then we ever read the storm The Ferrari use off zero 24 Then the way Good. So this should that rare a very, um is good too in they grow from Cyril to three. If we put the vory bro here they're very off eggs. Here we have ex full toh for two or three four like times before it's for two. Bull for justice. Yes, your four Dimas 64. Because for for a three for 3 64 press seven times four is 2028. We put zero here. 00 Here is roast saurus minus zero, then D y good. So this is good too. If we decorated this warren away. First picture tow away is that this is constant. Bribed by owner. Why here? We have to Ex key skipped you for so this very is good too. 16 256 over three. Press 20 Date Time Z and to burn off. All right, because this is constant is why so things wise? Three zero 23 So if we put three here scoot three here If we put zero, Here's to ABC wrote this is Goto two 56 press three times e 28 Use your grip geta and you get I'm Jake If I wrote the question of everywhere Yeah, this is yes for excess Creative process seven, this is collect on DDE. Yes, this is also correct. Good for Biggs Cube. Dorothy, worry off. Bex is too is not saving. Sorry it's not for this is too good. So to press two cube is eight. So this is started. Toe over three, not 64. Sorry. 32. That's two over three. There's no four here. Here is seven times to this is 14 14. Sorry for Mystic. He was taken. There is this one. So here, this is good too. So that's it too. Over. City Press 14. Time Z. Why? Because the chamber of this constant things ways Why then? Three zero. Which is good too. People Delivery of three here we have three times. Three times this one Justice Goto. That's it too. Then here we have 14 times three is 42 minus zero. Which is good too. 74 Nice. So the unit this is a unit off. Very have to put a unit of very cube. Didn't Cube? Maybe you're sent matter cubes Like this. Thank you. This is the answer

So to find the volume off. Is it in this questions? It is too, son. Ex cost. Why over the region off our which is bounded by zero and by over two for the X and zero and bio four for the why so to find the volume, we'll have to integrate over the region off our in over the state on DDE. So for this over that area, So for this week and integrate two ex first. And then why so for the why we have the boundaries off Pi over four and zero. And for the aches we have the boundaries off by over two angina. So it doesn't matter which one you integrate to first are the X or y. I just choose for now to integrate two ex first and they do why. All right, I can see that I can separate my variables so I can take the two and the costs. Why out off this? Integral? Because I'm only integrating two eggs so I can treat the cause. Why as a constant. So from the I'll have the integral off by over four, and I have so I can put the two completely and totally in front, off the whole thing to take it out. It's a common factor, a constant grow then. So I put the two in front. They not have more boundaries off. Why and I take out the cost. Why? And so now I can only integrate to eggs for what's left. So I only need to integrate sign off eggs. So the integral of sine off eggs is negative course off eggs and my boundaries day on by over two and zero. So let's substitute all of this in the then I will still have the two. I just want to substitute a ll the boundaries in. So I keep this for now and I will have so it's negative and cause off by over 20 minus and cause off. Zero is one. So it becomes negative one. And from this we have right. So this ends. This simplifies to one only. So I'm left with the integral off course. Why? Which is sign why over the boundaries by over four and zero getting on this side. So it's two times. And if I put by over four into sign, it gives me squared off to over two, putting zero into that issue zero. So this will simply file to this squared off too

We want to find the volume of the region bounded above by the surface. Z is equal to two side of x co sign of X m below by the rectangle where X ranges from 0 to 2 pi half and wiring is from zero to pi fourth in the X y plane. Now the equation we're gonna use is that the volume is equal to the double integral over a region are of our function f of X y d A. And one of the nice things that we can actually do since notice weaken right. This equation here, as the product of Onley functions that depend on X and only functions that depend on why, as well as our intervals over here do not depend on each other. They're just Constance. We could write this as a multiplication of two intervals. So I'm gonna make this as the integral from zero to pi half of to sign of X dx and then times the integral from zero to pi fourth of co sign of Why de y. And it's not too hard to actually show. This is the case when these functions are continuous and we have those other two restrictions because these are really just Constance with respect to each other. So let's go ahead and jump to this, though Now we can integrate each. So integrating sine gives us negative co sign. So that should be negative to co sign X and we're evaluating from zero to high half. And then we're going to evaluate this other one or integrate that, which is going to give us a sign of why ended. Evaluate from zero to pi for now, co sign of pie half zero. And then we'd have minus negative, too. So that becomes positive to co sign of zero is one. So that's just too. And then over here when we plug in pie Fourth, we should get route to over two. And when we plug him zero, we get zero. So these twos cancel out, and we're just gonna be left with the square root of two or the volume underneath that curve


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