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Meon value theorer And clnikcza" AfC ! fxx) Xt L]Volume3x1 X=0, X"2 Y=Solid...

Question

Meon value theorer And clnikcza" AfC ! fxx) Xt L]Volume3x1 X=0, X"2 Y=Solid

Meon value theorer And clnikcza" AfC ! fxx) Xt L] Volume 3x1 X=0, X"2 Y= Solid



Answers

Sketch the solid whose volume is given by the iterated integral.

$$\int_{0}^{2} \int_{-1}^{1}(2+x+2 y) d y d x$$

In this problem considered the iterated integral double integral from minus 1 to 1 and 0 to 16 minus two x minus D by D by D. X. Now the objective is to sketch the solid whose volume is even but this I traded integral. Now let f of x y is equals toe six miners to X minus three y then from I treated integral the function. I hope X Y is integral over direct angle are is equals toe X Y minus one less than equal to x less than equal to one and zero less than equal to why less than equal to one. So sketch the region off the surface. F of X is equal to six minus two x minus three by order a tangled by using the maples affair off. First, activate the package in maple by giving the maple food Now the maple input will be implicit. Blood Really? That equals toe six minus two X minus three y comma exit was toe minus toe comma Y equals minus two. That equals toe minus four 16. So this will be maple output here. This is X axis. This is why excess and this is Zahra, Texas. So that's the solution

Having this problem. We have toe sketch the reason of integration, which is 0 to 3 zero toe tu minus two. Expert three one minus one by three times X minus one by two James Away do I d. X and then we have toe sketch the solid also whose volume is given by this in trigger. So we know that these are the limits for why so why is equals to zero to why is equal to minus who works? Do it by three on these other limits for eggs so excessive calls to zero and X equals 23 I'll be laughing these equation a graph paper to sketch the region off integration. So this is why access on this one is X axis. Now this line the presented equation X equals 23 On this lane it represents the equation y equals two to minus two weeks by three on This is why calls to zero. That means X Texas On here X equals 20 That means the y axis. So the region of integration would with this region this regional presidents don't read enough indication now do we have were brought a solid. So for that I will be sketching this holiday. So this is the Y axis on. This one is Z axis on. This is X axis and this solid represents the volume of this integration. So I hope you got the problem. Thank you.

In this problem here we have discussed the solid whose volume is given by created Tintagel. It is double integral from 0 to 2 and 0 to 3 x squared plus y square D by and D x. So this is the figure of the given interested integral. Here we have this as X axis. This is by axis and this is the nexus. That's the solution.

Right. So with this interval, we like to see what volume and what solid it represents. So, um, the first thing we like to do is look at our order of integration, and we know that I said it x y x and y. So it doesn't really matter in this site for this problem, but in other problems, it does matter, because when you're looking at your balance, you need to know what but your region. So this is saying is that inter over our and in this case are just 0 to 1 both Texan. Why? So if this was like they were to, it would definitely matter which which order here and you're bringing in. So what we'd like to do now is look at what this into grand represents. So, um, we can think of this as a phone is being like the volume and their surface and the surface is Z equals four minus X minus two y. Now, um, this represents plane in that changes with both X and y. So it's gonna be like something like this, some plain that looks like this in the first quadrant. So, um, what we like to do now is find where what these points are in this plane. So, um, we think of a few points. If we say that Z is a function of X and y, we could say that had had the f of 00 for the origin and X Y space. You see, you just get for and, um, we think we don't think of where and points are. So are in points are a 01 in every direction. So we know f of better fortified after 10 Enough of 01 So this is just, um so four minus one or three and then four minus two to so we know three points already. Um, and now the three d space. So then we want to also have the 4th 1 f of 11 which is just four minus one months, two or one. So we have four points. 004 103 Syria 12 and 111 So then we can go ahead and graph this in three D space where we're gonna call this X. That's why in the sea, So then we have eyes Years or a four here, right with Red 004 up here. So this is the X axis. So this would be 10 three. Ah, 012 And why? And then, um, 111 So when one wants here and then we know that since this is a all you want to this is a volume under this this plane here, Um, it's going to be something like, um, things like this so that we're gonna have the square project the square underneath that have projected under. And so our volume is a sort of prism, but, um, it's not actually prison. It's just this plane surface I love, uh, and then with a square base.


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