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Find the volume generated by revolving the regions bounded by the given curves about the $y$ -axis. Use the indicated method in each case.$$y=8-x^{3}, x=0, y=0 \quad \text { (shells) }$$

Calculus 2 / BC

Chapter 26

Applications of Integration

Section 3

Volumes by Integration

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In mathematics, integratio…

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In grammar, determiners ar…

02:03

Use the method of cylindri…

01:13

03:00

Let R be the region bounde…

04:07

02:04

Use the shell method to fi…

02:21

04:47

02:56

00:42

Find the volumes of the so…

Even if you don't know what this craft looks like, uh y equals 8 -1 cubed. Like I know the Y intercept will have eight in there And I also know there's going to be an x intercept at two. Um and those are really the only things I care about Because they also give me the bounds of x equals zero and it's got to come down and we're doing shells around the y axis. So all I have to do is worry about doing two pi From 0 to 2. The radius of each one of these cylinders is X. And then the height of each one of these is 8- execute. So what I would do is distribute the X and find the integral at the same time because it would be eight X. That when I add one to the exponent and divide by my new X. When I get 4 85 by two is four. Now this is X to the fourth, that when I add one to the exponent and then divide by the new exponent a little trickier. But either way you can go and plug in your bounce, you know four times 2 squared is 16 And then 2 to the 5th 2, 48, 1632. And if you plugged in zero, you would just get a pair of zeros That you have to multiply by two pi at the end. So 16 -32. This 16 -30 to 50 Is a fraction of 48 fists. That when I double that answer because there's a two in front, I get 96 pi over five. Now, it should be a right answer.

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