OIm 5 $ 9 71 V the QUESTION 4 " eolume . 2X= the solid generated revolving the region bounded by the given lines pue 1Click 1 1 Submit 1 1Find About the x- the...

Find the volume of the solid generated by revolving each region about the given axis.

The region in the first quadrant bounded above by the curve $y=x^{2},$ below by the $x$-axis, and on the right by the line $x=1$ about the line $x=-1$

So this curve why equals negative? X cube can be re written as X equals negative wide the 1/3. And, um, we need to know where the two curves intersect. And that happens at why equals one. So that will be our bounced. Integration abounds of integration. And also, of course, and zero, they are other about immigrations. We get 0 0 to 1 of, um pi r squared. So now, um, applied that big r squared minus little r squared. Um, And then, of course, the, uh, variable of integration is why, Cindy? Why? Um and big r is, um negative. No negative. Why? To the 1/3 um, minus negative to this, Plus to you and little R off the smaller radius is just three, you know, negative one. Minus negative too. Is our started. You know, it's negative, but three, because we only care about the length. So this integral is going to be the integral from zero 21 of pi times big R squared, minus little R squared, which is going to B. Why? To 2/3 power. Oh, miners too wide to the 1/3 power plus three integrated. Do you? Why and so this. It's going to be pie. We anti differentiate with plug in the numbers. PI times 3/5 minus two times 3/4 plus three That's our answer.

So in this problem, we need to find a volume of rotation. We're gonna have to use the washer method because our axes of rotation, which is X equals negative too does not touch our region. Uh, everywhere, even anywhere in this case. So our region is bounded right here. This is the Lion X equals negative one. This curve is why equals negative X to the third. So our region is right in here. It's were limited to the second quadrant. We're also bad about the X axis then as well as X equals negative one. And because we're going to be rotating around a vertical line, we want to use horizontal rectangles here, first of all, finding our endpoints. Since we started the X axis, we're gonna start at zero here. This point here to see where that is. What has set are two guys equals a negative one equals negative X to the third divide by negative one. We get one equals X to the third to take the key brute, and we get that X equals one is the next place those to cross. So those were gonna be our boundaries on our region. Now, here we're using the washer method, like I said, because our region is not touching the axis of rotation. So our formula is going to be pi times the integral from a to B. So the beginning of our interval to the end of it, Judas founder endpoints for of the outer radius squared minus the inner radius squared. Do you? Why? Because we're rotating around a vertical line. Reason horizontal rectangles So are out already is. And our iterated Asare outer radius is how far the furthest edge of our region is from the axis of rotation eso that's going to be negative. Que brute of why? Because that's what we get. If we rearrange this equation here, why equals negative extra? The third becomes X equals negative key brood of why just by dividing by negative one and taking the cube root minus negative two. So that's how far this far edge from the Cuba curve is from. X equals negative, too. And our inner radius is going to be the closer edge of our region. The X equals negative one to negative choose. That's negative one minus negative, too, and I noticed weaken simplify both of those the outer radius is negative cube root of why plus two and the inner radius is just one. So we go to find our volume. Now we have pi times the integral from zero ones. That's the bottom in the top of our region of the outer radius squared. That's negative. The cube root of why plus two squared minus one squared de y with times the integral from 0 to 1 of well, multiply this out November negative. Um, que brood of why is negative? Why to the 1/3. So we square that we get negative. Why to the 2/3 Or that just becomes why to the 2/3 the negative will cancel out. Ah, Then we have minus four que brood of why plus four minus one de y like terms there V equals pi times the integral from 0 to 1 of why to the 2/3 let us four wide with 1/3 on the plus four minus one makes plus three. So we'll take our integral now Just rewriting That's we have it on this page. Take her interval We get pi times Why the 5/3 times 3/5 Because we divide by 5/3 of that. The same is multiplying by the reciprocal minus. This is gonna be why to the 4/3 now that's going to be the four that's already there. Times another 3/4. There's those forced Cancel out Ah, and then plus three. Why? From 0 to 1. So we get V equals high times. We plug in the one 3/5 minus three times one to the 4/3 that's just minus three plus three and then minus zero minus zero plus zero. We plug in zero G zeros for each term. So that's just going to be pi times 3/5 or three pi over five.

Let's do the cylindrical method. So I'm gonna draw an elemental cylinder here. So this is what the cylinder is gonna look like about why access this will be. The radius on the radius is actually gonna be equal to X, and the length off this cylinder would be to Rex minus X squared minus X. So my elemental volume deanie would be equal to two pi r l dx, which would come out to be two pipe times x times two x minus X squared minus X T x. So, in order to find the volume, I take the integral to pie in a girl from 0 to 1 that's times X minus X squid. The ex that integral comes out to be good to one over 12 and when at times it would want to pipe the volume is pie or six. So the volume would be pie or six for Part B. I have to do the same thing. But now my access off rotation is actually equal to X is equal to one. So again I'll do the cylindrical method. So here is my elemental cylinder. The radius would now absolutely be one minus X so Here's my radius. This would be one minus X and the left is still the same. Soul length is still two x minus X squared minus x So that's my length and the elemental volume Devi would people to two pi r l D s and that is actually put to two pi times one minus x times two x minus X squared minus X d x. So now, to find the volume, I take the integral to pie immigrant from 0 to 11 minus x times X minus. X squared DX. The integral is equal to pi over 12 and when at times it with two pipe, the volume is a player or six.

In this problem. We're trying to find him of all of you, of the solid phone, by retreating the bounded area in the graft bill about the white activists. No, since we're going to be doing X integration and were involved about vertical axis than it appears that we will be using the shell method. So let's first identify oh, samples the lender reading the equal to act under Hank equal to you Need to negative. You're wrong with being 42 times the inter, both from X equal zero flexible one one x time. You need to know the best square. Yeah, No one took danger, bro. We have negative one house dreams you need to the negative range thing. While you did it to one that would be have bring out the negative Bahnhof Hi times. I need to the negative born minus into zero that would be equal to one minus. One of it in temps way is beyond value