00:01
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 intersect the x -axis, and that have centers along this parabola.
00:25
So let's graph this parabola really quickly.
00:32
And if i consider this parabola as being the center of one of the circular cross -sections, let's put a center right here at 0 .6, just as a sample cross -section.
00:51
The circles are perpendicular to the x -axis and intersect 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, and one representative cross -section would look kind of like that.
01:28
I'm not drawing a circle, i'm drawing an ellipse because i'm trying to show it cutting into the plane.
01:34
So this is the radius of this circle.
01:40
Another representative slice could be here.
01:49
That's not the world's most beautiful circle, but oops.
01:57
So that could be another representative slice.
02:00
There could be another representative slice here.
02:03
So the centers of the slices are on the parabola.
02:07
The tip of the slices touches the x -axis because the circles intersect the x -axis.
02:13
And the overall shape is going to make this sort of croissant shape with circular cross sections.
02:20
It has a flat edge between negative 1 and 1 on the x -axis and has a curved edge that is twice as tall as the parabola given by the function.
02:37
I made that circle a little too big.
02:40
I'm sorry about that.
02:42
So this is the shape that we are trying to find the volume of.
02:47
If i look at it, at a circular cross section.
02:51
I'm going to pull one of these representative circles out.
02:53
So this is the cross section looking directly at the x -axis.
02:58
The radius of that circle is given by the y value, where y is equal to our function one -half of 1 minus x squared.
03:17
And that is what we are going to use to find the volume.
03:22
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.
03:46
And since they are perpendicular to the x -axis, the delta -x will give me the width of the slice.
03:56
So area times width is going to give me the volume of each individual slice.
04:00
And all of the volumes added together will give me the total volume of the shape.
04:10
So first thing i need to do is come up with an expression for the area.
04:20
The area of one individual circle is given to me by my formula, one -half of one minus x squared.
04:44
That's the radius.
04:47
Let me rewrite that.
05:00
So the area of one of the individual circles, the area of a circle is pi r squared.
05:09
But the radius of the circle we've already said is the y value.
05:14
That's given to me by one half times one minus x squared.
05:23
So taking that radius and squaring it and multiplying it by pi will give me the total area of the slice.
05:35
And then the volume of the slice is going to be the sum of all of those areas added together.
05:42
I can write that as an integral.
05:47
So the total volume is going to equal.
05:51
Pi is a constant.
05:53
I don't have to have it inside the integral.
05:58
My limits of integration are going from negative one to one.
06:07
And i am integrating, oh, one half squared is also a constant, so i can multiply that out front.
06:21
And i am integrating 1 minus x squared, squared with respect to x.
06:42
It doesn't like that limit of integration, so let me see if i can fix it.
06:50
That's better.
06:53
And it doesn't like my d, so let me see if i can fix that.
07:01
And it doesn't like my pi over four, so let me see if i can fix that...