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Use the method of cylindrical shells to find the …

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Problem 15 Easy Difficulty

Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the specified axis.
$$y=x^{4}, y=0, x=1 ; \quad \text { about } x=2$$

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Calculus 2 / BC

Calculus Early Transcendentals

Chapter 6

Applications of Integration

Section 3

Volumes by Cylindrical Shells

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Video Transcript

Hello. We were ready to welcome back. Okay, so we're in. Let's see. We're still in section 63 Right. Okay, so Chapter six, Section three. He Oh, Paige made a new 4. 45 Number 15 in Stuart's calculates with early transcendental sense. You have boobs, you have. Why you called Sex of the fourth, which is sort of parabolic. Looks sort of like a parabolic flat bottom. Okay, where the region is bounded by that Weichel zero, which is the X axis and X equals one. This we're going here. So here's the region for revolving. Right? Okay. Now, the thing is, though, that we're revolving it about the line. Why? Mexico's too using cylindrical shows. So if we set up a vertical strip and thickness of DX, it's like, this is the ex, the Heidi's ex, the fourth. But what's the radius? The radius is gonna be two minus sex, isn't it? Because when x zero all the way at the origin, that's two units long. All the way to expose, to write. But when Mexico's one when we hit exit goes one, that's only one in the room, and it's anything in between, like the next is the F That would be 1.5 minutes for the radiance. Right? So my remind statement is the volume of us one little slice influence has been thin. Slice is gonna be two pi times the radius rotation. So this is a conference of rotation in terms of height, of the of the shell, which is the function given in terms of thickness of the shell. And so we're gonna integrate that from 01 Right? So we covered that whole region was she did before. All right, so let's work that out. So the volume is two pi the integral Tu minus x X to the fourth DX from 01 So I guess we should distribute the extra fourth. Two extra fourth minus excellent. Fifth. I would integrate yet quantity the X. So integrate X before to get extra fifth over five. We still have that, too. In front minus integrate extra 50 at Exit 66 integrated from 01 one. Plan zero and then subtract those two values in that order. Now we're booking zero. That's just zeros out. So we're just gonna put in one Du pi times 2/5 minus one sixth right. All right. So we could come to non leaders here. We had to buy about 30th. Also to fifth, somebody by six. Over 6, 12 30th. Mine is also talking about 1 65/5. We got 5/30. Okay, she end up with two pi over No. Seven place setting. No, just wait. 675 two pi times 7. 30. That's what I'm trying to say. And the two or 30 reduces to 1/15. So it's 75 15 cubic units. Okay, But that was helpful, right? Have a good day, little Joe Maxilla. Bye bye. See you next time.

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