Problem 61 Hard Difficulty

Find the volume of the described solid $ S $.
The solid $ S $ is bounded by circles that are perpendicular to the x-axis, intersect the x-axis, and have centers on the parabola $ y = \frac{1}{2} (1 - x^2) $, $ -1 \le x \le 1 $.

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in this activity. We are being asked to find the volume of a solid that is described as having cross sections that are circles that are perpendicular to the X axis that intersects the X axis and that have centers along this parabola. Let's graphic this problem really quickly. Right. And I consider this parabola as being The centre of one of the circular cross sections. Let's put a center right here at .6, just as a sample cross section. Yeah, the circles are perpendicular to the X axis and intersects the X axis and have a center on the parabola, which means that the radius of the circle is the Y value of the parabola and the diameter of the circle is therefore twice the y value of the parabola. Right. And one representative cross section would look kind of like that. I'm not drawing a circle and drawing any lips because I'm trying to show it cutting into the plane. So this is the radius of this circle. Another representative slice could be here. That's not the world's most beautiful circle. But oops, so that could be another representative slice. There could be another representative slice here. So the centers of the slices are on the parabola. The tip of the slices touches the X axis because the circles intersect the X axis and the overall shape is going to make this sort of crescent shape with circular cross sections that has a flat edge between negative one and one on the X axis and has a curved edge that is twice as tall as the parable, given by the function I made that circle a little too pig. I'm sorry about that. Mhm. So this is the shape that we are trying to find the volume of. If I look at a circular cross section I'm going to pull one of these representative circles out. So this is the cross section. Looking directly at the X axis. The radius of that circle is given by the Y value. Where why is equal to our function? One half of one minus X square. Yeah and that is what we are going to use to find the volume. So the volume is going to be, where am I going to write this? The volume is going to be the sum of the areas of each individual circle times the width of each individual circle. And since they are perpendicular to the X axis, the delta X will give me the width of the slice. So area times with is going to give me the volume of each individual slice and all of this. All of the volumes added together will give me the total volume of the shape. So first thing I need to do is come up with an expression for the area. The area of one individual circle is given to me by my formula one half of one minus X squared. Okay, that's the radius. Let me rewrite that. So the area of one of the individual circles. The area of a circle is pi r squared but the radius of the circle we've already said is the y value that's given to me by one half times one minus x squared. So taking that radius and squaring it and multiplying it by pie will give me the total area of the slice and then the volume of the slice is going to be the sum of all of those areas added together. I can write that as an integral. So the total volume is going to equal. Hi is a constant, I don't have to have it inside the integral. My limits of integration are going from negative 1-1 do you think? Yeah and I am integrating But 1/2 square. It is also yes a constant. So I can multiply that out front and I am integrating one minus X squared squared with respect to X. Doesn't like that a matter of integration. So let me see if I can fix it, that's better and it doesn't like my d so let me see if I can fix that. Yeah, no it doesn't like my pi over four so let me see if I can fix that. Mhm. So this is my integral, going to go ahead and just turned it into math. This is where it comes from, I'm going to go ahead and just turn that into mouth and then to begin integrating this first I'm going to expand this one minus x squared squared. So that will give me that volume equals pi over four times The integral of negative 1-1 of one squared is one negative X squared minus x squared is a negative two X squared. Make that two. Yeah and X squared squared Is a positive X to the 4th, negative times negative is positive. Yeah, so that's rewriting everything a little bit. It doesn't like this, it thinks that that is and eyes and let me make that clearer. Yeah, there we go. And one more change that I can make is this is being integrated from negative 12 positive one. But it's symmetrical so I can make this a little bit easier by integrating from 0-1 and then doubling it. So I'm going to change this 20 and I'm going to double it. So instead of being divided by four it will be divided by two. That's right. Well it doesn't like the two for some reason. There we go. Yeah, yeah, so this one might be my integration and this is easy because it's all just power rule for integration. So that means that taking the anti derivative I get the volume is going to equal pi over two times the anti derivative of one with respect to X is just X. The anti derivative of X squared with respect to X is one third X to the 3rd power And the anti derivative of X to the 4th power is 1/5 Of X to the 5th power. And I am Integrating that from zero 21. Right. Didn't like that. So let me rewrite and to integrate from 0 to 1. I plug the one in and then I plugged the zero and I subtract the values. So doing that I get that. My volume. Is it going to equal pi over two times? Yeah plugging one into X. Is one. Plugging one into X cubed is also one. So I get minus two thirds. Plugging one into X to the fifth. Power is also one. So I get 1/5 and plugging zero into all of this gives me just 00 plus zero plus zero is just zero. So I don't have to even subtract that. I can just deal with the one that's why using the symmetry made this easier and then simplifying The equals pi over two. Multiplying everything by 15. I get 15 minus 10 plus three. Yeah All divided by 15. Mhm. Okay. Okay and That means that my volume equals 15 -10 is five plus three is 88 divided by two is four. So I got four pi divided by 15. It doesn't like my writing four. Hi divided by 15. Mhm. Okay and that is my total volume