5

Point) Consider the solid under the graph 0f >v" above the disk 22 + <a? Mhere(a) Set up the integral to find Ine volume O1 the solldInstructions" P...

Question

Point) Consider the solid under the graph 0f >v" above the disk 22 + <a? Mhere(a) Set up the integral to find Ine volume O1 the solldInstructions" Please enter the integrand in the first answer box, typing theta for = Depending the order of integration you choose , enter dr and dtneta either order into the second and third answer boxes with only one Or Or @theta each box, Then; enter the limits of integration;L"IOO0(D) Evaluate the integral and find the volume. Your answer w

point) Consider the solid under the graph 0f > v" above the disk 22 + <a? Mhere (a) Set up the integral to find Ine volume O1 the solld Instructions" Please enter the integrand in the first answer box, typing theta for = Depending the order of integration you choose , enter dr and dtneta either order into the second and third answer boxes with only one Or Or @theta each box, Then; enter the limits of integration; L"IOO0 (D) Evaluate the integral and find the volume. Your answer will be in terms 0l 0 Volume V = (C) Whal does the volume approach as l 00?



Answers

$[\mathrm{T}]$ Use a CAS to graph the solid whose volume is given by the iterated integral in spherical coordinates
$$
\int_{0}^{2 \pi} \int_{3 \pi / 4}^{\pi / 4} \int_{0}^{1} \rho^{2} \sin \varphi d \rho d \varphi d \theta
$$
Find the volume $V$ of the solid. Round your answer to three decimal places.

Okay, so we're having a girl from the George Inn in a girl from you have exquisite yourself in a girl from zero to one plus discover bit of experts. Why, 30 z d. Y. And the executed others into a non checked in and we get this is approximately equal to 0.6.

In question were given and tempering to co I'm doesn't to. One woman is 1 to 1 for minus square. The x d. Why? It was trying to draw the figure here you were. This will function inside. Here we read the ah rub a light on. It has the high acquittal for here. It's going down here. Then we have this ago on the place. Now and then, we have considered, uh, X is innocent. Accident. Why on dizzy? And we only considered xbg minus one and one. So the minus warm is here, and ones on one is near and minus one on the other side. And why was from 0 to 1, someone's here so it would draw on the vertical line up? Yeah. So this will be the region that we're looking for here. It would be like this, Andi. It would be on the other side. Yeah. So this will be the ricin. Will it fall? Okay, so this will be the bottom. We're looking from here, so I look at it. It's kind of hard to miss. Stayed here. No, in Italy. Very. This problem will do this department. The girl. So from 0 to 1. We came the outer for now, for the inner we have extremely the variable and yeah, not just dentition is from the month 1 to 1. Therefore, here inside this one is even. And this one is also the everything is even under forward. You have the two times now we will have the entire if don't wear have been recruited. Do that is the number minutes one joint have is there to one for minus X squared minus y squared the x no. And then we get equity the rooms to 12 on the entirely rift in this one Echo Jetta for X minus. Expect to have a Jew when it's quite square. X evil it and desert you one g y. Then we get the rooms or 212 And now it will be the one inside that actually getting good. You two for minus one of the Jew minus y square. And then do you? Why? And it wasn't if I've indicative rooms that you won four months 1/2 so good you It would be, uh, with him stew as well. So it minus one. Good. You seven minus two y square do you? Why? And then we will get equal to entirely riveted this one in calculus. Want seven Y minus two Weber 3/3. Even though its energy to one didn't get Nico Judah seven minus two and three. Then we get you got your 21 minus two. So you could you 1903.

Okay. What we want to do is we want to find the volume of a solid generated um about the Y axis of the following region. And the first thing we need to do is to kind of get that sketch developed. Um and just to kind of help us out. And so the first thing we do is schedule region and so this is one, this is zero um and appear at one. And so we know, oops that went off at smart, didn't it? Um We know it looks something like this, that is why you go to exclude. And so we know it's why equal to zero. So it's going to be the X. Axis and X equal to one. So it's gonna be this region right in here that we're evolving about the Y axis. Okay. And of course we're gonna be doing um the shell method. So let's get our representative rectangle drawn. So let's just put a rectangle right here. Um We know the with that rectangle is delta X. The heights of that rectangle is why equal is X squared. And the height of the rectangle of course is is to this X squared. And then this distance to the center of the right tangle is X. Right? Um and so we know that the changing the volume is actually equal to two pi X. So it's gonna be this distance right here, two pi um X times the height time sir with that rectangle which is about two pi X cubed delta X. Right? So here is our representative volume and so now the volume is equal to the integral and we're going from 0 to 1 and this will be two pi X. Hope that should be a cubed dx and there we have them so now we can kind of integrate and find that volume. Um bring out the two pi and then of course this is going to be 1/4 x to the fourth. And we're gonna evaluate at zero and one. So this is gonna be pi over two times one minus zero. So this is just gonna be pi over two. So there is a volume right there.

Okay. What we want to do is we want to go ahead and find the volume. Using the shell method. Uh The solid that is generated by the region bounded by X. Equal to screw to want Y plus one. Why I called the four X equals zero Y. Zero about the X. Axis. Um So the very first thing we should do is is draw that graph, right? And so we're in quickly drawl um the region that we're interested in, so if y is zero, X. Is one, so we're right here. Um If why is one, why is one then X is over here at two. And so it's kind of kind of looks something like of course I miss my mark, something like this. Um So that is what that's gonna look like. I just quickly did a couple points in my head um to 34 This is why I go to four right here and we're doing X equals zero, which is the Y axis and why go zero which is the X. Axis. So it's this region bright in here that we're interested in and we're evolving it about the X. Axis. So don't forget our rectangle for doing the shell method and we're evolving about the horizontal, the rectangles horizontal. So then the second thing we need to do is to draw that representative rectangle. So let's just draw right tangle in here. And so um the width of the rectangle is delta Y. And the distance from the axis rotation is why? And the height of the rectangle or the length of the rectangle is this square root of why? Plus one. Right? Um And so the change in volume is equal to two pi the distance from the axis of rotation times the height that that rectangle times the width of the rectangle. So our volume is the integral and ry is going from 0 to 4. Um And I probably need to put the two pie in here to let's let's change that over. So the volume is equal to two pi where I bring that out 024 And this actually is going to be why to the three halves plus. Why do you why? And there we haven't so now I know how to integrate this. Right? So this is two pi times 2/5. Why did the fifth five halves? Sorry about that? Um Plus one half Y squared. And we're going to evaluate at four and then again at zero. So this will be equal to two pi. This will actually be two to the fifth which is 32 times another two which is 64/5 plus. And then this will be mhm. 16/2 which is eight. Um So if you put it over a common denominator it's going to be two oh eight pi over five and there is our volume of that solid


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