use-the-method-of-cylindrical-shells-to-find-the-volume-generated-by-rotating-the-region-bounded-b-5

Question

Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the y-axis.

$ y = x^2 $ , $ y = 6x - 2x^2 $

Amrita Bhasin · Answer

We know that if X is to y squared than what we have is why is one again reuse the method of cylindrical shells, which means our our is to minus y and r h is gonna be to minus two. Why square, in other words, to minus explore using why cats do wise the two minus two. Why square? Which means plugging in what we know we have to pie times two minus y times two times one minus y squared. Now we need to look at this in terms of the integral from 0 to 1 to minus y times one minus y squared, Do you? Why, I would recommend using the power rule me integrate It makes a lot easier because now what we&#x27;re doing is we&#x27;re increasing the exponents by one. And then we are dividing by the new exponents just like us. Which means plugging in. We have four pi times one to the fourth, which is just one minus two times one cubed, which is just again one times two on the top. One squared over two plus two times one, which gives us four pi times what&#x27;s on the inside. So we end up with 13 pie over three

Amrita Bhasin · Answer

we know using the method of Shlain Driscoll shells. This means we have a radius and a height. Our radius is two plus. Why are height is gonna be one minus y squared? Which means if we&#x27;re plugging into the formula to pi, times are bounds from negative one toe. One were multiplying radius times, height. So what this means is we have two plus y comes one minus y squared. It&#x27;s pretty straight forward in terms of plugging in now, I would highly recommend simple Find this before taking the integral negative. Why cubed? Minus two. Why squared? Plus why Plus to do why now we integrate. We&#x27;re gonna be using the power rule, which means increased the exponent by one divide by the new exponents, which means we have fee is 16 pie divided by three.

Amrita Bhasin · Answer

In order to use the method of cylindrical shells, we first know that we need to find the intersections. So if access that equal to four x minus x squared solved and we end up with acts Times X minus three is you remember we&#x27;re trying to solve for acts forgetting intersection points because thes are actually gonna be our bounce. Now remember the general formula for volume as two pi times into girl from A to B, which is the earthly We just figured that out with our intersection points are age, times D backs greatest times Hey, essentially OK, plug in Everything we know are times h. Now we know that with me integrate, we&#x27;re gonna be using the powerful, which means increased the exponent by one divide by the new Acciona. Okay, Now we know we&#x27;re gonna be plugging n when we plug in zero. We actually just end up with zero Simple five. We end up with 27 pi divided by two

Amrita Bhasin · Answer

we know that the general formula for volume is gonna be our bounds from A to B which you just cases wanted to to pi times our radius acts times our height acts cute times, DX This is the general formula. We just plugged directly at pull out our constant, which we know is to pie Simplify x times x cubed just simply x the fourth. Now we know we&#x27;re gonna be using the power rule to integrate, which means we increase the experiment by one. So to the fifth divide by the new exponents, which means now we can plug in so plug in our bounds. Constant still on the outside. Work. Stay. Now we have two pi times 31 divided by five which gives us V is 62 pi divided by five

Amrita Bhasin · Answer

using the method of cylindrical shells. We know that we have to set equal toe one post y minus two squared because we have to find the intersection points for why those are gonna be our to bounce. So we have to do this. So? So for why, again? We&#x27;re trying to get why equals, once we have factored, we know we end up with our wise one and three. Good. Those are two bound. Now we have this. We know that if ours why and the height is gonna be two minus one plus y minus two squared. Then we need to simplify the height also to make it easier to plug. And otherwise it&#x27;s gonna be really messy. This is negative. Y squared. Plus four. Why last three? Ok, good. That&#x27;s something we can now put into the integral. Now, remember, the formula for interval is to pied times bounds, which in this case is from 1 to 3 of two pi r h do you want are two pilots That is a Constance. We already put it out there. So R h d y wide times negative y squared plus four. Why minus three d y okay Now, when we integrate, we use the power method, which means we increase the expert at 51 and then we divide by the new Expo in This is the easiest way to integrate. We know we have our bounds from 1 to 3, which means now we can actually plug in our batons. We have two of them, which means we have to do top minus bottom, and near of them are zero, which means we actually should plug in the bounds. Oftentimes, when we have zero is one of the bounds, it ends up being that zero plugged in. Just get zero so weakened neglected. However, in this case, we can&#x27;t because we have one on the board. I&#x27;m not zero, which means we actually have to plug it in and see what we get. We end up with two pi times four minus 4/3 which gives us 16 divided by three pie. When you do the algebra

Wen Zheng · Answer

in order to use the method of cylindrical shells. The first thing we need to do is find intersection points. In other words, set our two values X squared and six x minus two X squared off of X and G of X, equal to each other. So for acts to better intersection points which will then become our bounds, says you can see this problem requires that we figure out our own bounds. Now, remember, the formula for volume is two pi times bounds from A to B integral or Veitch, which means radius times, height of D of X. So now, given this, we can actually planning what we know, as we just specified are too. Intersection points were zero in two. Therefore, lowest one goes in the bottom. Hi, Simon goes in the top acts time 66 months, three x squared again Our times h times d backs. Okay, pull out our constance three times to a sex sex pies pulled out on the outside. Use the power rule. Increase the exponents by one divide by the new exponents were the point now where we can actually plug in our bounds. You can probably tell from looking at this, that if we plug in zero, we literally just end up with zero. So you don&#x27;t have to write that as long as you recognize the fact that the zero doesn&#x27;t affect the problem in this context, simplify this further. We end up with eight pi.

Problem

Let $ V $ be the volume of the solid obtained by …

Problem 7 Medium Difficulty

Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the y-axis.

$ y = x^2 $ , $ y = 6x - 2x^2 $

Answer

8$\pi$

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