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PointsSCALCET8 6.3.003Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the Y-axis 5,7,Suby...

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PointsSCALCET8 6.3.003Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the Y-axis 5,7,Subynnil Ansior2/2 pomnisSCALCET8 6.3.013Use the methed Cylinameal enella Iina tho volumc Vol the colld obtalned by rotating the region bounded by the qiver curves about tha 5)2 ,SkelcKconon[ypicil thell

points SCALCET8 6.3.003 Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the Y-axis 5,7, Subynnil Ansior 2/2 pomnis SCALCET8 6.3.013 Use the methed Cylinameal enella Iina tho volumc Vol the colld obtalned by rotating the region bounded by the qiver curves about tha 5)2 , Skelc Kconon [ypicil thell



Answers

Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the specified axis.
$$y=x^{2}, y=2-x^{2} ; \quad$ about $x=1$$

Problem is used my third of its cylindrical show house to find that the warrior generated by rotating the region. Bonnie Device Cruise Other than given access Why is he goto Sign I acts were true. Why's the serial access between zero and one? But Y axis through the weekend jobs graph is as follows This is from zero to one and the graph of wise he goto sign packs where to so region's thiss part. And when we need to rotate this region about why access and find the volume So use cylindrical shells My third, we can see Watch him while we get caught You shoot high into girl from zero one hams packs arms Relax. What is this Call Sign? Hi. Ike's over too. The axe. Now for this problem, we can use integration my parts So the formula is And to grow from a to B you be from the axe. It's cultural. You have sway from a to B minus into girl from a to B You from house Leave Jax now for our problem. We collide. You sick too Axe and a promise to who assigned hi ice over two. Then you prom has one and B is two. Sure. Fine times, Signe. Hi, I over two now. Just WeII if he cut your high times, buy this farm owner to see you. This is our two lower high times. I sign I over to from zero to one on minus integral off your prom comes we. So this is to over Pi signed. Hi. Ike's over two from zero to one, Jax. Now this is he called too. This is Chico. Chico. Too high times. Hi. Over too. Replying one and zero to ACS. This is a want you one and then minus two over a pie. I am so integral of sign packs over to this is equal to make typical signs of his ass. Sign us to our high sign. I Iike older two from zero to one. This is the country True high times Hi! Tour behind us. Oh, our high squire arms consign pirate with zero and Kasai in here is one. The answer is four minus Ain't over, Pi. This is our answer

We want to find a volume that is generated by rotating the region bounded by the curve. Why equals uh E to the negative X squared. And uh the lines Y equals zero, X equals zero And Ux x equals one. Uh So the region that is bounded by this curb. And these lines, that region is going to be rotated around the y axis. And we want to use the method of cylindrical shells to find the volume generated to get a better idea idea of what this region looks like. I have a uh this curve and these three lines graft on the dez most graphing calculator. All right, so here's the portion of e. Actually it's craft incorrectly. Let me fix this. We raised to the negative X squared. There we go. Okay. Uh So here's the portion of the curve Y equals E to the negative X square between ex uh equals zero and X equals one. And of course this is the line Y equals zero. So we're looking at this little region in here, let's blow it up a little bit. We're looking at this region in here uh is going to be rotated around this y axis. So we're going to take this region and spin it around the y axis. And we want to use the method of cylindrical shells to find that volume. Well volume. Using a cylindrical shells is going to equal to times pi time she entered role. Looking at the uh region. Again, we're going to integrate from X zero up to x is one uh X times our function. Why? Which is E. To the negative X squared D. X. So once we calculate this definite integral uh That will be the volume when this region is rotated around the y axis. And this is using the method of cylindrical shells. Well to evaluate this integral. Uh First um let's go back to let's just go off to the side here. What is the integral of X. He to the negative X squared D. X. If we use U substitution let U equal negative X squared. Then D. U. D. X Would be -2 X. Uh so d'You would be -2 X. D. X. Let's get a space here. So I wanna rewrite this integral in terms of use now this will be each of you. So in a girl. Alright each negative X squared. Um Each a negative X squared will be E. To do you. No X. D. X. Yeah is equal to -1/2. You. Uh So if we want to put A D. U. In here we need negative two X. Dx. So if we want a negative two X. D. X. So that we can write negative two X. D. X. As D. U. That means I want to put a negative too negative two. Let me write this negative two has two times this. But if I'm time seeing this by negative two we have to balance it by multiplying out here by negative one half. Negative one half times negative two is positive one. So I'm not changing anything by multiplying by one. But the E. To the negative X squared is E. To the U. Uh Since negative X squared is my U. E. To the negative X squared is E. To the U. The negative two X times dx That's my D. U. And then of course I have the ties by negative 1/2. So this integral is really uh the integral of uh X. E. To the negative X squared dx. This is the hardest part of the problem. Not that it's hard but when I find this integral I'm basically finding the integral of X. Each negative X squared dx. So I'll know what the anti derivative of X eaten negative X squared is. Once I know the anti derivative, then I just had to evaluate it between the limits of X X one and x zero. So once I calculate this I'm going to get an expression uh in terms of you and then I'll substitute back in what U. Equals. So then I'll have uh an expression in terms of X. Which will basically be the anti derivative of X. Each negative x squared dx. So Uh integral v. To you. That's an easy one. The enemy derivative of EUU. Is each to you. So this is equal to negative 1/2 E. To do you Which is really equal to negative 1/2. Uh Since you is negative X squared negative one half. Each of the eu is negative one half E. To the negative X squared. So not our final answer. But this is the anti derivative of X. Each negative X squared. If you took the derivative of this, if you want to check your work, if you take the derivative of negative 1/2 times E. To the negative X squared, you will get X E to the negative X square. So negative one half E. To the negative X square is the anti derivative of this. So running out of space, I'm going to rewrite this uh volume formula. Uh Using cylindrical shells rotating around the y axis. I'm going to rewrite this formula down here and we're actually almost done. Okay. The volume in that region is rotated uh around the y axis. Using the method of cylindrical shells is equal to two pi Times the integral from 0 to 1. Uh E it was X. E. To the negative X squared D X. So this will be the volume of the region. Once it's rotated around the y axis. Now, the anti derivative of X. E. To the negative x square is uh this expression right here in the red box. So this volume is simply going to be two pi times The anti derivative. The negative 1/2 E. To the negative X squared uh evaluated between X is one And X0. So basically we have to uh calculate the value of this definite integral. So we take it the anti derivative and we evaluated at one and then we subtract the same anti derivative evaluated at zero. Now just to make life a little easier, we could do the two times in one half and we can do that and you'll still have this negative. So let's let's take this one negative one half. That's times and all this. And let's move it out here just to make things a little bit easier. So two pi times negative one half will be negative pi. So we really just have to take negative by in times of by E to the negative X squared Evaluated between one and 0. Well that's going to equal negative pi times while each of the negative X squared when X is one is E. To the negative one. Then we have to subtract e two negative x squared evaluated at zero, which would be E. To the negative zero square T. 20 which is one. So uh multiplying negative pi by each of these uh negative pi times negative one is pie negative pi times E. To the negative one is a negative pi. Each a negative one but each a negative one is 1/8. So we can write this as minus pi Over each of the one or 8. So there is our volume when we rotated that region around the y axis. Using the method of cylindrical shells

First thing we should do is we should draw the function just a rough sketch. And then we know that if this is our so called shaded region, then we can consider perhaps that we have a cylindrical shell that looks a little something like this. As you can see, this is the 0.1 comer one as specified in the problem. Now, remember, the formula for volume is from A to B two pi r h d x. Remember, ours acts h is ex the 1/3 or the cube root of X. Now that we've given this, we can pull into the volume formula what we have, which means we can we can pull out the constant which is to pie. We have a bound instead of a to B. We have 01 We have acts, which is our times X to the 1/3 of the cube root of ax exponents are easier to manage because now we're gonna be using the power rule, which means we're gonna be increasing the exponents by one and then dividing by the new exponents, which means we now have six pi over 73 times to a six. As you can see, we're divided by seven times one minus zero, which gives us six pi divided by seven

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'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'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'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.


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