find-the-volume-generated-by-revolving-the-regions-bounded-by-the-given-curves-about-the-y-axis-u-29

Question

Find the volume generated by revolving the regions bounded by the given curves about the $y$ -axis. Use the indicated method in each case.
$$x^{2}+4 y^{2}=4 \quad 	ext { (quadrant I), (disks) }$$

Gregory Higby · Accepted Answer

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&#x27;m taking that for the lower bound to be Y equals zero. Yeah. And what we don&#x27;t know is where it&#x27;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&#x27;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&#x27;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&#x27;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.

Russell Arnold · Answer

So to find the bounds of immigration, we must must set the curves equal, come to find the intersections. So we have four minus at our bounds of integration. Four minus X squared equals two minus X. And this implies we could use the quadratic formula, but it can be seen more directly that X equals two or one. Um, so the bounds of integration are going to be 1 to 2 of pi r squared rpai, big R squared, minus little r squared. And, um, things are square and nice Little r squared. Um, it&#x27;s all for it&#x27;s going to be integrated. The axe, um, and big are you is equal to you. X minus two. And little R is just equal to four minus x squared because of axis that were evolving around. Then we can plug this in this into this formula. So we get in a girl from 1 to 2. Uh, well, pull out the sky, then we have four minus for X. That&#x27;s nine x squared. Come on. Minus x 1/4. The ex. Um and so this is V. So then this is going to equal for X minus two x squared, plus three x cubed minus X to the fifth over five. And then the grounders evaluation, of course. For two on one and altogether this equals 24 months. 32 5th plus our sorry minus five minus 1/5. And that&#x27;s our answer.

Kevin Maritato · Answer

So this problem We&#x27;re finding a volume of rotation. We have to do it using the washer method because we have a gap and at least part of our region. Between the edge of the region and the Axis rotation, which here is the X axis. So are two functions that give the bounds of our region. Here we have this line, which is why equals two minus X. And then we have the parabola, which is why equals four minus X squared. Now I&#x27;ve already marked on here where they crossed. We should find that first. So we have four minus X squared equals two minus x Go to solve. This will move the X squared over here and the floor. Also, we get zero equals X squared minus X minus two. If we factor that, we get X minus two and X plus one. So that&#x27;s where we get those intersection points that I marked on the graph. We have X equals two and we have X equals negative one. That&#x27;s that&#x27;s where there&#x27;s intersect, meaning that our region is the upper part on the graph right here we&#x27;re gonna be adding up are a bunch of washers who is ready, I or, um who&#x27;s slice is really are given by these rectangles here between the two curves. So with washing method, we have an outer radius and an inner radius. The outer radius is further from the axis of rotation in this case, the X axis. So our outer radius is going to be the top curve here, which is four minus X squared, and our inner radius will be the one that&#x27;s closer to the X axis. That&#x27;s going to be the two minus X right, and our volume is given by pi times, the integral from the beginning of the interval to the end of the outer radius squared minus the inner radius squared D. X, in this case, because we&#x27;re rotating around the X axis. So our volume here will be pi times integral from negative 1 to 2 of the outer radius squared. So that&#x27;s four minus X squared squared minus the inner radius square. That&#x27;s two minus X squared. D X will expand those We have pie times integral from negative 1 to 2 of 16 minus eight x squared plus extra. The fourth minus four plus for X minus X squared D X now combining like terms we get V equals pi times the interval from negative 1 to 2 of X to the fourth minus nine X squared plus four X plus 12 DX. We&#x27;ll integrate that only integrate. We get X to the fifth over five from integrating the extra. The fourth minus nine extra the third over three from integrating the nine x squared plus four x squared over two from integrating the four x plus 12 x from integrating the 12 evaluated from negative 1 to 2. And now we have to plug those bounds in. So we have the equals pi times looking into we get 32/5 minus nine times eight to that 72/3 plus four times four. So that&#x27;s 16 over to plus 12 times juice. That&#x27;s 24 minus. If you plug in negative one negative 1/5 plus 9/3 plus four over to minus 12. This is 32 5th 72 divided by three is 24 so that&#x27;s minus 24. That&#x27;s plus eight plus 24. Over here we get plus 1/5 that&#x27;s minus three, minus two plus 12. Okay. And If you put everything together here, get pi times 33 5th plus 15 or that&#x27;s high times 33 5th plus 75/5, which is 108 I over five, and that&#x27;s our volume.

J Hardin · Answer

here we&#x27;d like to find the volume obtained by rotating the region bounded by the curves sign and co sign about the line Y equals one. So here&#x27;s a rough sketch of the graph down here and read. This is the graph of sign from Zero power for and then in blue we have the Cho Sang graph. The green line is the axis of rotation. This is the horizontal line Y equals one And then the red and the blue graphs up top those air The reflections After we were rotated this area down here around the line Y equals one So we see after that we do is a revolution around the line y equals one we have ah, this hole in the middle of our solid. So that means that our cross sections which is indicated by this washer right here we&#x27;LL have holes So we have washers. Oh, so we have a formula for the volume in this case also, we obtained the volume by letting the washers move in the ex direction. So our follow ume will be in terms of X. So we know the volume is the integral. See you in a power before times the area the washer, which is pi times the larger radius squared minus the smaller radio square. So we need to find formulas for big R and little r. So it&#x27;s relying the picture for this. So looking at the picture here, we want a large radius which is the distance from the center all the way out to the end point. So this is a big R. So one way to get there is to take this entire distance from the X axis two, the horizontal green line, which is one. And then if we subtract this distance right here from zero to the red graph, then informally, we have equality here. So we take the whole distance, which is one, and we subtract off the distance from X all the way up to the red. And we&#x27;re left over with this big r. So we have big our peoples, the entire line which is one minus this. Why value here? But since this Y values on the red side graph, we could replace why with signings. So that gives us big r and then for a little R the same idea. We want the smaller line right here. This line segments a little radius, but we can get there by taking the entire line segments or the one and then subtracting the distance from zero all the way up into the blue graph. And then we&#x27;re left over with lous little r So we have little r equals one minus y. But this same reasoning is before. But this time the y goes all the weapons for the blue graph. So this time why is co sign? So that&#x27;s how we find big. Our little are in this case now we&#x27;re ready to evaluate the integral. So first we should simplify as much as we can. It&#x27;s quite implode that pie. So inside we have one minus to sign, plus sign square minus one. Then we have a minus minus to co sign and then minus coastlines where so we could simplify this a bit. So let&#x27;s go ahead and we see that we could cancel the ones and we can apply Formulas should be a X there for science. Where and co sign squared. So use those double angle formulas with a pie in a girl&#x27;s ear apart before minus two. Sign eggs, plus then this becomes one minus CO sex co sign of two X over too. We still have this to co sign and then we have a minus and this becomes one plus coastline to X over to. So we&#x27;LL need more room here. So it&#x27;s goingto new new page so we can go ahead and cancel the one half in the minus one half in the previous page. So picking up where we left off, we have negative to sign X minus coastline to X over, too, to co sign X And then we have a new way says we have to times two of these. Let&#x27;s go ahead and just put those together. Erase this. So this is just after simple simplifying the previous expression. Now we could integrate into these so we have to co sign X minus signed two eggs over too, plus two cynics and our end points or zero power before. So it&#x27;s Quentin plug in these end points, so we plug in power for first. So co sign power for is rude to over too sign of pirates who is one? And then we have another two times room to over, too. When we plug in zero Ko san a zero was one. So we have two times one sign of zero zero. So these two terms go away and were we can simplify to get our final answer. Hi. We could combine these radicals to obtain too radical too. And we have minus a half minus two, which is negative. Five over two. So this is our volume, and that&#x27;s your final answer.

Russell Arnold · Answer

Okay, so here said typical um disc method alarm volume solutions. So the bounds of integration are given negative pi over four two pi over four and the you&#x27;re a pi r squared r is second squared x So come the anti guerrilla who seek out squared acts is tension tax. So we get a pi times tension by over four minus attention of negative pi over four. So that is just going to be, um, to pry, because each of the is this one of those negative ones survive.

Adnan Gill · Answer

So why is he going to seek and off X Which means why square we&#x27;re people too. Secret square defects. The volume would be cool to any grilled tie. Why square the X from zero to fight over four or volume is equal to We&#x27;re gonna put the pie in front so five times integral from zero to high over four seeking squared x dx The integral of Seacon Square X is changing decks so we have V is equal to high and change index with the limits off integration from zero to high over four so t would be cool to pi times changing off pie or four minus changing off zero and Kenya compile for is one changing of zero is zero so the water is equal to pi.

Problem

Find the volume generated by revolving the region…

Problem 24 Easy Difficulty

Find the volume generated by revolving the regions bounded by the given curves about the $y$ -axis. Use the indicated method in each case.
$$x^{2}+4 y^{2}=4 \quad \text { (quadrant I), (disks) }$$

Answer

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