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Problem 53

Use the region $R$ with the indicated boundaries to evaluate each double integral.

$$\iint_{R} x^{3} y d y d x ; \quad R$ bounded by $y=x^{2}, y=2 x$$

Answer

$\frac{16}{3}$

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## Discussion

## Video Transcript

we will take it to every way to turbo Integral. Oh, this is double integral and distant. Min x x x Cube times Why do you Why your friends? We, uh this our testament are is bonded by just two impressions while you go to extra spare and do I you go toe to eggs you're to everywhere to disturb your integral would like to find the bones. Those founders roar, burn for eggs Throw a bone for why oppa bony for X and proponent For what? So to find this barons beds just a sketch or those to function. You see, this is where I could to excess square. This is a quadratic function where we heard for why I could why go to x x squared? You see that if we heard value X equal to minus, let's say minus two. Why would it be good too home when you were a nerve crazy Implement us one. Why's goto What when we have zero x go to zero wise going toe there When he heard why goto one exclude no one. Why's Goto Why when we have a Y kowtow to excluded for and sword when you have three Here. We have a name. I want a boat. Why? Good toe minus What? No, For a seat. Other boats. This one wouldn't. When we have wash, you could too. Toe eggs seem very key. Organic? Half if we have minus two This We have minus four. If we have remind us one. Yeah, minus one. Why would it be cool to minus two if we have 07 times 00 Why would we go to zero when you have for one? Never have a tool. When you have tool, you have it for when you are three. You ever have six in the so called soul. So? So let's try to sketch those two functions. Maybe her. All right. This is okay. Okay. This is very from pecs. This is why here. Sorry. Here. May story was direct story to where? There. Now it doesn't work. Okay. Yeah. Make it. Does pro fish from my as good as I come when I can extend it here. This book. So Okay, let's catch this function. Size touched. But here? Yeah, we have one. Have to. You have three. You have one. Have tool. You have to Yeah, we have one. I have to. You have three, you know, for Try to think like this. This is a way. This is it. Picks. So you have one? Yeah, we have to hear. Huh? Three have minus one. They have minus tool. You have my last three. Yeah, we have one. I have to. They have three her for good. So just get this first function that make fuss function with brute. When you have negative tour, we have full. You know that negative toe before is some point here You, When we heard of negative four on here is the one means that's the point here for her to zero. We have a point here. If we heard of one and we have the one is We have a point. He we have to We have full Bihar for profit again. Here to form I po int he So you have to draw it away. So when it was try to sketch this function, This is a quadratic function course. Remember that You have to use this how to measure the unit. So is like this look so fresh. I think that's right. Did where? Good? Yeah, or function repressed to zero. This is zero. And then you go up good. The 2nd 1 Let's give It's the girl for bread cake. When you have minus four minus two, we have minus for somewhere. Hey, if we have minus one is minus two somewhere he free her of zero zero. We have one is two one is too. Okay, when we have two were full here when you have to. We have six here, so we'll have a Ryan, which is like, Okay, straight line. So means that what we need the boners we needs is this one. This is too fu. Okay, now we have to find the intersection because the section is this warm friend. How the bones. So in this case, for this terminus, where X is from zero to two and weigh a why is from 0 to 4. So this is the R. This is Ranger for our so means that here, we're gonna have into sugar. He got through into rule French rates by exit from zero two before his integrate. We're integrating with respect to why is Israel 24 receiver where the intersection is 24 to 4 for this function? so on. Duh. What's the function? Function is X cube things. Why, then the Why the ex? Yeah. Three are here This cape So we can integrate this sworn with respect to west Know that X cube is constant So we ever were injured. Great way were tricked Great story with respect to away which is good too Rehearse in chambers from 0 to 2. Okay, this is X cubed. I'm from around the place. But wise remember that if we have why and take off White is goingto y que y squared over too. Then we put very offer why which is good to four and zero The awful eggs. This is good too for too poor two is 16 so 16 over two is eight x cube Inject The Christian is ex cured. Yes, X cube. Listen take from 0 to 2 we put here is 16 16 over two is eight x cubed d'Or for pics. This is going to if we integrate eggs for integrates this with respected toe eggs That is X cube would be eight. Remember that we put we're happy if we have extra Poor Kay is k press one off a k press want. So it's the ex too poor for exit a pair for over four extra bar for Is that you then tour here. And one here is the ex to board next. 444 This is two X to perform two to par for is 16. You have to times X to poor form. Is it due to poor form? But he is not one into serial minus zero. Too poor for So this is 16 then The answer is 30. True. This is there. Yes, sir, for this program.

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