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Sct up the integration necessary to find the volume of the solid that is formed by the graphs of the equations.2x + 3y+42 = 12,* = 0,y=0,2 =0.Set Up only, Do Not Ev...

Question

Sct up the integration necessary to find the volume of the solid that is formed by the graphs of the equations.2x + 3y+42 = 12,* = 0,y=0,2 =0.Set Up only, Do Not Evaluate:Use a Double Integral6.) Usc & Triple Integral

Sct up the integration necessary to find the volume of the solid that is formed by the graphs of the equations. 2x + 3y+42 = 12,* = 0,y=0,2 =0. Set Up only, Do Not Evaluate: Use a Double Integral 6.) Usc & Triple Integral



Answers

Draw the solid region whose volume is given by the following double integrals. Then find the volume of the solid. $$\int_{0}^{6} \int_{1}^{2} 10 d y d x$$

All right. So this video, we're not going to actually integrate this we're going to do is we're going to draw a geometric representation and use that to find the volume. And so we're going to do first is we're going Teoh, map out this on the X Y plane. So if we look at this x Y plane, we see that ex Im going to do in bread goes from zero Teoh, Let's just say six. You hear that? Why is going from one to to this is to this one. Um, what we know is this right here The difference here is one. The difference here is six. And so we put this like on a rectangular prism. We're going to see that is this is in the X direction. We have six, and then we have one from this. And then we have this 10 here, and that's going to serve. Has the depth of this rectangular prison that we've created that's going to be 10 and we're trying to find the volume. We multiply 10 by one by six and we're going to get 60. All right thing guess which were watching. Feel free to go back and pause and rewatch it. That's something you need

So in this problem we want Thio. Find the volume of solid and close by to Paraiba Lloyds eso. To do this, we're going to want to use the volumes. It's DX, dy DZ um and we have our different values that were given. We know that X is from negative to to to, so we'll do an integral from negative to to and we see that Z is from negative square root of four minus X squared to positive square root of four minus X squared. So our next in a role will be and negative square root four minus X squared on the upper bound will be. It's positive and then lastly, we'll have an integral of X squared from X squared plus C squared to a minus x where minus this word. And this all comes from determining the bounds of the solid that we have. So now that we have this, we want Thio evaluated, it's gonna be g y then BC, then the X. So let's start with the first in a role. Um, and that's obviously just going to be why evaluated at thes bounds and what will end up getting as a result is eight minus two times X spurred plus z squared. And then, um when we evaluate this with the Z squared or the dizzy right here, then we'll evaluate it. And we could also use polar coordinates to change this up. So if we do use Polar Burnitz, this will just be from 0 to 2, and this will be from 0 to 2 pi. So once we do this, we have to change it on the inside, and that will be a minus two r squared. And that will need to be our grd Seita talking. And we'll keep this in practice. Ease. Now that we have this, um, we see that this value is not changing even if we change the coordinates that we're using. So then this will just be multiplied through. So we'll get in our cube and this will be a are. And then when we take ah, we'll move d theta to be right here and then we'll multiply them. So then the d theta will obviously give us two pi and then this one right here will be our for our squared minus one half part of the fourth, evaluated at two and then evaluated at zero. That's gonna use two pi times 16 minus eight, which will then give us eight pi times to which will be 16 pie. And what you'll see is that this value has not changed it all over the course of the entire problem, which tells us that we've been calculating it correctly. Um, and it also shows that as long as you are allowed to use a calculating tool, you might as well save some time and start off just by determining the integral. And then once you do that, your calculation will be correct.

In the question, we have to use the triple integral to find the volume of a given solid, the tetrahedron and closed by the coordinate plane And the plane. Two weeks plus y plus that is close to four. Now, moving towards the solution here, why is 000, two weeks plus zero plus zero will be equal to four. That sp so the points are too common. Zero comma zero. Similarly your points for Q will be zero comma for comma zero and four are will be zero comma zero, comma four. No the it can be written as two weeks plus Y plus zero is close to four. Why will be 4 -2 weeks? So integrating from 0 to 2 and integrating from 0 to 4 -2 weeks day by the X. So now moving further That is equal to 4 -2 weeks -Y. So integrating from 0 to 2 into integrating from 0 to 4 minus two weeks into integrating from 0 to 4 minus two weeks minus Y D said d by d X. Now we will be solving this integral and solving or integrating first with respect to X. You will get integration from 0 to 2 and two. Integration from 0 to 4 minus two weeks. Four minus two weeks minus why the why the X. Now integrating with respect to why you will get Integration from 0 to 2. Four into four minus two weeks minus two weeks into four minus two weeks minus one by two into four minus two weeks. Full square deeks. Now moving my back integrating with respect to works. You will get uh sorry solving the previous equation, you will get integration from 0 to 22 X squared minus a tax plus eight dx. Now integrating this you will get two by three x q minus four x square plus eight X. And limit going from 0 to 2 which will be equal to 16 by three. And so this will be our answer. Thank you.

In English. They are Lose the devil into lyrical. Find the volume of this solar which is enclosed by y squared very close to zero and extras ethical stolen. Now moving towards the solution one will be given by double integration over the lesion. A affects my the in this problem explicit. There is a whole school one and Britain as medical school. One minor sex. So we will be given by the willing figuration north original one minus x Do you this integral and Britain as integration from minus 131 into integration from by square to 11 minus six dx dy like firstly, we will be solving the interim cigarettes Keeping their ordering funeral is it is really good X minus excess whereby limit going from my sweat one. Do you like? It will be quicker integration from minus 11 of one to minus twice where place fly physical four By holding this instigators one of one to minus one upon sleepless one by them that they ate in. And this will with finalized for the human question. Thank you


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