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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.$$x^{2}+4 y^{2}=4 \quad \text { (quadrant I), (disks) }$$

Calculus 2 / BC

Chapter 26

Applications of Integration

Section 3

Volumes by Integration

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So since we want this we want to do high revolving around the X. Axis with the function squared in terms of why? Uh So they also tell you that they just want quadrant one. So I'm taking that for the lower bound to be Y equals zero. Yeah. And what we don't know is where it's going to be a Y intercept. So quick answer though is you could take the function and give you the X squared Plus four, y squared uh equals four. And plug in zero for X. Because that's going to give your X intercepts. So you can divide the four over. And then when you square one get positive and negative one. But we just want the positive version. And like I said, I want things in terms of why? Um So what I would probably do is sulfur. Why? Bye. I guess technically speaking just subtracting X squared over first and then divide by four and then square root in that piece. It gives me whatever that shape is an ellipse. But what's nice about this is the square root will cancel out with a square. So at this point I can evaluate this integral pretty easily. Just X. You add one to your extreme and then multiply by the reciprocal 1 4th Times 1/2 is 1/8. And do that from 0 to 1. So as you're looking at the way up, I maybe rewrite one as 8 dates minus 1/8. And as you plug in zeros you just get a couple zeros in there. Anyway, that comes out to be 7/8. So seven pi over eight.

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