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Point) Find tne volume of the solid enckced by (he paraboklds < = 4 (*and = = 50 4(2 +3)...

Question

Point) Find tne volume of the solid enckced by (he paraboklds < = 4 (*and = = 50 4(2 +3)

point) Find tne volume of the solid enckced by (he paraboklds < = 4 (* and = = 50 4(2 +3)



Answers

Find the volume of the solid that lies inside both of the spheres $$+4 x-2 y+4 z+5=0$$ and $z^{2}=4$

Okay, so for this problem, we're re evaluating problem 30. Sorry, 41. So we're looking at the equations. X equals Y squared and X equals y, and they're revolving around X equals negative one. We're still going to look at the interval from 0 to 1. So what I want to do is I want to evaluate the inner and outer. So this one's going to be my outer radio. So it's going to be one plus white. This one will be my inner radi I, which will be one plus y squared. And since we're comparing two different equations, we're obviously gonna be using the washer method. So it's gonna be pi into girl from 0 to 1. Gonna take that out or radio I, which is going to be one plus why and square it and subtract one plus y squared all of that squared. So now what I'm going to dio is I'm going to square everything out. So one plus two y plus y squared minus one minus two y squared minus white of the fourth. And then I'm going to start simplifying everything down one more time so my ones will cancel. Then I have to y Why? Squared minus two y squared. So minus y squared and then minus. Why did the fourth? So this is going to be pie of Why squared minus one third. Why? To the third minus 1/5. Why did the fifth from 1 to 0? Yeah, so this is going to be pie. And then, of course, one minus one third, minus 1/5 and then minus zero. So then my final answer will be seven pi over 15.

Okay, So for this problem were bounded by why equals four X And why equals X squared? And in between, um, where X is in between zero and four. So for this one, I'm going to use the washer method and the volume, um, formula is gonna be pi times the integral between a and B of f of X squared minus G f x squared DX. So this is going to be pie between zero and four of four x squared minus X squared squared DX. So this is now going to be pie between 40 and four of 16 x squared minus X to the fourth. And so this is going to be pie of eight. I'm sorry. 16 3rd x squared X to the third, minus 1/5 X to the fifth between zero and four. And so this is going to be, um, pie and then 1024 over three minus 1024 over five and then, of course, minus zero. So this is going to be 2048 pi over 15

Volume of between With the region between these the function and y equals three. So sketch what this looks like first here. So the square root of 25 minus x squared. It's going to look like this by equals three. She made this tree line, Michael's three here. And so the values where this is happening at and they intersect, it's going to be here a negative for Over here at four. Okay. So want to find this region rotated the X axis here, which means that we're going to use the wash your method. So with this in mind, let's go ahead and take Girl from -4- four. And inside I am going to be squared of 25 minus X squared squared. There's three script Simplifying that out. We have 25 minutes x squared minus nine. Since the last 16 minus X squared -4-4. Spy that's going to be equal to 16 X minus x cubed or three -4- four. That that's pie. But that's basically going to be equal to four times 16 which is 64 And 64 or three. And then I'm just going to multiply this by two because negative four double negative makes it the same thing. That's what we had times pi. So the answer is going to be 26/3. That's why.

So we want to find the volume with these regions created by these equations. First thing you do is get, it turns wise we've got one equals X squared. I want to hear me if y equals four x. Here, the two of this year, the they say we will meet at Cereal and then four. So as 40 zero square to zero and four squared and four test for both equals 16. It's gonna be our region here. So we're going to basically use the washer method because take a look at X squared And we have four x. Right? You've got a gap right below it here. Well, this region because it's going to go about the excesses here. Yeah, I can't below it here. So one on top is the forex Suite 2 4 x. Yes squared minus X squared squared. So simplifying here, he got 16 squared. Had sex to the 4th. The Girl from 0- four. Spy. So dealt with Spy Times 16 execute over three In 1654 or five. Take a reserve for. So that's going to give us pi times And that's 16 times four cubed over three. It's 4 to the 5th over five. That's going to equal 2048 Over 15 times pi.


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