Find the volume of the described solid $ S $.
The base of $ S $ is a circular disk with radius $ r $. Parallel cross sections perpendicular to the base are squares.
Applications of Integration
it seems like you guys fell asleep so I might want were given a solid S and rest to find the volume of this solid, we're told that the base of S. Is a circular disk with a radius. Little are that parallel cross sections perpendicular to the base? Are squares look well. But is it so the equation of the semicircle above the X axis? This is why equals the positive square root of little. R squared minus x squared. And the equation of the semicircle below the X axis is Y equals negative square root of R squared minus x squared. These bounds the base and therefore we have at the side length of the bottom of the square, which I'll call S. This is our upper value of why? Which is the square root of R squared minus X squared minus the lower value of why? Which is the opposite of the square root of r squared minus x squared which is two times the square root of r squared minus x squared. Right? Yeah, all turns out so this is the length uh side length of the square pointing out of the circle. Yeah. Now define the volume. We should find the area of one of these squares as a function of X. Well, I was well this is the area of a square with a side length and so it's going to be the side length two times the square root of r squared minus x squared squared, which is four times r squared minus x squared Israel to. And now we have the volume using the cross section interpretation. This is the integral from X equals negative art positive our of our area A X of the squares pointing out of the circle, D X. This is the same as four times the integral from negative are positive. Are of r squared minus x squared. The X. I can imagine this is a pretty simple integral to evaluate but I'll walk through since we do have some and all the variables in this. So this is equal to take the pussy on four times R squared x minus well mr Niva. And this is X. To the just letting us know that Science third power over three from X equals negative are two positive are and if you plug in well we get four times To r cubed over three. It's named that wasn't there bitch? It rode a horse naked lady which is equal to she really she's not very good doing china. He rides around yes minus four times negative. Two. R cubed over three. You can see it. This is equal to is sloppy eight plus eight or 16. R cubed over three. Good job. Really good.