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Point) Find the volume formed by rotating the region enclosed by: x =llyad y = x withy > 0 about the Y-axis_ Volume...

Question

Point) Find the volume formed by rotating the region enclosed by: x =llyad y = x withy > 0 about the Y-axis_ Volume

point) Find the volume formed by rotating the region enclosed by: x =llyad y = x withy > 0 about the Y-axis_ Volume



Answers

Find the volume obtained by rotating the region bounded by the curves about the given axis.

$ y = \sin^2 x $ , $ y = 0 $ , $ 0 \le x \le \pi $ ; about the x-axis

This problem is from chapter seven section to problem number sixty one in the book Calculus Early Transcendental. Lt's a condition by James Door here we like to find the volume obtained by rotating the region bounded by the curves. Why equal sign X y equals zero xon between pie over to and pie. And we like to rotate this about the x axis over here on the right, we have a graph of y equals sine X for X between pyro too and pie. This is in blue. We also have in red y equals zero and then we have the bounds. X equals pi over to this is in green. This is the line vertical line X equals power too. And then our upper bound for exes over here at pie so we could see that the region bounded by all of these curves Is the area inside here? This object will seek a fin So it's commented, called Eurasian buy Some letter was calling our and moreover, rotating this are this region about the X axis. So some notation that used might see here is something like that to let us know that we're rotating this thing around the X axis so we could go out and actually draw what the solid would look like after the revolution after the rotation. So after we do so we get an object that looks like this, and then we can go ahead and make it look like a solid. So this thing with Wade you right here this our disk This is a cross section and we see that it's a washer, so let eh be the washing area. And we know that our formula for the volume the is that integral of the area of the washer. So it will be the inner world, eh? So we need to be more specific here and our problem. The is a washer. So the area will be pi r square. That's our into grand. And this problem we obtained the volume when this washer moves in the extraction. So we should be integrating with respect to X. And in doing so, we should use the bounds pirates who and pie for limits of integration. Also, we need to find our which is the radius of the washer. So are the washer radius. So let's go to our picture here. Let this be our starting point. So we see that this is the center of the washer, the center of the disk. And if we move straight up, that's a radius right there. And this is just some why value. So our equals Why we don't want to use why in the general, Because we're in a gated with respect eggs. So we use the fact that here this why, Val, you were on the blue curve. So why is given by signing books? It would be better to use this because now we're in terms of X. So are integral becomes less claw that pie. It's a constant power to the pie. And then we have signed squared of X. To evaluate this inaugural it'LL be best to use an identity a triggered a metric identity for sign square. We can write this as Science Square is one minus coastline of two x, all divided by two. And since two was just the constant, we could also pull that to a outside of the integral. And we can evaluate each of these in a girl's and a girl. One is X, and the integral of coastline of two weeks is signed two eggs over too. Where am I help you to use a U substitution here. If you can't see this for this integral, you could try and use up. U equals two X. Can I look and let's not forget our end points piracy Tobi. So it's good and plug these in for X if we plug in pie first. So let's do that. So we have pie minus sign of two pi over too. Then we plug in pi over two for X minus. Sign a pie over to you. Separate this from our scratch work on the side. So sign of too high from the unit circle. We know that zero sign of pies also zero. So we're left over with hi over too. And then pie minus pi over too. It's also a pie over too. And so we got our final answer for the volume. Pi squared over four and there's our answer

Mhm. Okay. So for this question because our lower function is just why equals zero. Nothing uh Nothing else, Not some complicated function. And because we're rotating about the X axis. Um this this is going to be relatively easy. Our volume is just going to be um high times the integral from our lower boundary upper bound of the function squared. So in our case um mm hmm. Up our lower bound zero upper bounds pie. And f of X is sine squared X. And that's all squared to take the integral of this were well, when it when we see sine squared uh we can rewrite that in terms of coast to X. Um and that's we do that because that's easier to integrate. So basically what we're going to have is the is this mm ah well, in place of science squared determine identity. Which is that science sine squared X. Is equal to half of um Okay, Yeah, half of 1- Coast to X. Um So yeah, basically because of that, what we can do to make things simpler. Simple is just a sub that in there. So we're gonna have half of one minus coast two X squared. Okay. And what we're going to do is okay, before we even take the definite integral, we can just we can start by taking just the indefinite integral. So we're going to have mhm. We're gonna have that we're just gonna be ready. We're only taking this without the pie. And without the bounds. And when we do that we square the half we get Yeah. Mhm We get one quarter and we can take that constant multiple outside and then we have one minus coast to ex uh squared. Um Now we to integrate this we're gonna expand that square so we're going to have the first term square which is one minus two times first term. Second term is gonna be minus two coast to X. And then plus that second term squared. So coast squared two X. Okay. And integrating that. Well the first two parts are easy enough. You're gonna have integral one is X integral of negative coast to X. It's gonna be minus sign to X. And then but in that last part is going to be um we're gonna have to do that separately because it takes a few it takes a few steps. Basically we're going to use a different identity which is um which is that coast squared data is equal to have of one plus coast to X. So we're going to have here is one quarter of Yeah all that. Yeah. Yeah. And um plus the integral of half of one plus. Okay so our theater is two X. 02 times two X. Is going to be four X. Okay. Yeah. Can take that constant multiple out and mhm that means this is going to be this mhm. Yeah. Yeah mm basically we can put half and then integral of one that's going to be X. Again right and integral of coast four X. Will integral of Costa Science. So we're gonna have a plus 1/4 sign four X. And now okay, I'm just gonna keep it consistent. I'm gonna put square brackets around brackets on the outside. But yeah, now we can expand, expand this half to these two. And what we have is 1/4 times a X minus sign to X plus uh half X plus 1/8. Sign for X. Uh Combined those like terms for X. And we have one quarter of 3/2 X. My assign two X Plus 1/8 for X. So what was the what were we even finding here while we were finding um we were finding this integral, we're finding the definite the indefinite integral for this without this pie. And we're gonna go back um I'm just gonna move this down so we can work with it. Uh This is what our volume is supposed to be equal to. Yeah, okay, so RV is going to be equal to this and we just found that this is going to be um You got to this. Um But of course uh uh I just didn't bother reading an arbitrary constant by the way, like plus C. Because it's gonna get cancelled out anyway. But if the question was asked for an indefinite integral, you need an arbitrary, you would need that arbitrary constant. But yeah, basically it's going to be equal to this for the balance. So pi and zero. So now we got to do is uh okay we can keep that that constant, multiple potential and ford which power four can keep all that is the outside. But now we're just going to sub um the upper bound into this part inside the brackets and then the lower bound into that and subtract. So we're gonna have 3/2 times pi minus sine of two pi Plus 1/8 Times a sign of four pi. And we're gonna subtract and sub in the lower bound. So. Uh huh. 3/2 times zero minus sine of two times zero plus 1/8 of sign of four times zero. That second part this is zero Sign of 0, 0 And sign of zero is zero. That whole part is just zero. And for this first part sign of two pi zero sign of four pi zero. So those are all those two terms are also equal to zero. What we're left with is Um pi over four Times 3/2 times pi. When you multiply all that. Our final answer is three Pi squared over eight. What

The region is rotated around the X axis. Find the volume bounded by. Why couldn't do act square? Why equal to X and X equal to zero X equal to one. So here from the graph, you can see that we will slice the region, slice the region perpendicular to the XX. Is giving circular dicks of thickness delta X. So here you can see that from the growth in a radius is also in equal to ex choir and outer areas is also out equal to X. So the radius of disk is square off. All the ideas negative squared off in a radius. So here volume of slice, approximate by time. Square off. Also out negative by time. All sub in and here we have to put Squire dime delta X. So now we have to put the value of outer and inner radius. So here we get bye time X squared negative. Act two. The power four time delta X. Swimming over the bounds of the curve volume of solid form, total volume approximate sigma Five time acts quite negative. Act two the powerful time delta X as thickness of slice tends to zero. So here we get equal to definite intrigue of five time Acts to the power to negative act to the powerful time there's the X. So here you can write like that dx because here as thickness of slides tends to zero so he repaired D. X from 0 to 1. No, you can write like that by time, definitely intriguing off acts choir negative actually powerful time dx from 0 to 1. Now here we have to find anti derivative wolf ex choir negative. Actually about four is XQ upon three negative X five extra about 5.5. So we put here bye crimes X cube upon three negative X five upon five from 0 to 1. So now here probably is one. So we put here X equal to one. So here we get bye time one to the power three upon three negative one to the power five upon five. Naga. Do lower value is zero. So you put here X equal to zero. So here we get zero to the power three upon three, negative zero to the power five upon five. So now we simplify this and here we get bye time one upon three negative one upon five. So here we get to buy upon 15. So it is our volume by bounded region. So it is our final answer.

So for this question specifically because the second function, the lower function is just like 00 or the X axis and it's rotated around the X axis. We can we can easily just use this formula for volume. Um If those two weren't the case, it be a different story. But that means that basically to start finding the volume for this question, we're going to need to just first find this indefinite integral. The indefinite integral of the function squared. So we're going to start by taking the integral of of a science squared X. Yeah. Yeah. And to do this, you've probably done it before. We just use that identity. That sorry, that identity I wrote in green. Just so bad in place of it because that's easier to integrate. So we're taking the integral of one. No, sorry, we're taking uh the integral of half Of 1- Coast to X. Mhm. You can always take an arbitrary sorry, take a constant multiple out outside like that. And uh integrating we get 1/2 half of sign two X. Mhm. Let's see. So basically our volume based on this formula is going to be um Okay, high times the integral of from well the bounds are Hi over 2, 2 pi so yeah. Mhm. Um for science squared X. That's what it's going to be. And for that integral, that's just gonna be um half of X minus one. Over two times sine two X. For the upper and lower bounds. So over two for the lower end pie for upper and now we're just gonna stop those into Well we can of course keep this constant multiple by half or power to outside of it. And we're just gonna sub pie into this X minus half signed two X. Yeah sorry Yeah we're going to be um subbing the upper bound high into X minus half signed two X. And then subtracting sub the lower bound five or two into X minus half. Sine two X. So basically that means we're gonna have okay half sorry pi times half as I said it's gonna be pi over two. But yeah we're gonna have we're gonna sub first pie we're gonna have pi minus half of sign two pi. They were going to subtract and we're gonna sub the lower bound by over two pi over two minus half of sign. What's two times pi over two. That's pie. Mhm. Now we just need to calculate what all this is. So I'm gonna start by okay so signed two pi that's zero. So this part is just zero. You know cross that out and uh sign pie that is also zero. Cross that out. And and then what does that leave us with? We're going to just uh expand this this negative to bracket. So really just it's gonna be minus pi over two. So we have high over so we have that first pi over two and then uh times pi minus pi over two. Of course pi minus pi over two is by over two. So. Mhm. Multiply the fractions, and our final answer is that the volume is high squared over four. Thank you.


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