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Numerade Educator

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Problem 49 Hard Difficulty

Find the volume of the described solid $ S $.
A cap of a sphere with radius $ r $ and height $ h $.

Answer

$\frac{1}{3} \pi h^{2}(3 r-h)$

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Video Transcript

So were given a solid S. And we're asked to find the volume of this solid S. Is a cap of a sphere bon Jovi. Yeah with the radius R. Wilson and a height. Uh huh. Each there's some like and deep in yes. Now in order to um find the volume of this cap, we'll draw a circle however we'll draw it in a certain way. So set up our X. And Y axes and then we'll have a circle like this. The distance from the origin to this point on the right is going to be the height of the cap H. And so God works. The radius of the circle is going to be Yeah. Yeah. Words. Yeah. A little r We got the pork scores, murdering sucks. Yeah. All right. No I didn't even finish. But you said you're really now since the whole radius is our follows at the center of this circle is at the point let's see this is our, sorry negative are plus H. Zero. You know c. three. And therefore the equation for a circle is X. That's kind of what I wanted minus H minus R squared plus Y squared equals the radius R squared those better what? Look you can't say hard ours. Here is your anger. Anger reached a hard art. Then they got James, Earl jones to play Darth vader because he's black blue. I am now imagine rotating this across the X axis. But we end up with is our cap and we end up with disks use discs have a radius which is why equals the positive square root of r squared minus x minus h minus r squared. It was it was and shylock. So great they would play console. This is one way to approach it. I never noticed, let's say. And so the volume V by the disk method is given by the pi times the integral from x equals 02 H. Of our radius which is the square root of r squared minus x minus h minus R squared all squared dx one. So simplifying this is pi times the integral from zero to H. Of r squared minus x minus h minus or squared dx Mhm. If you take anti derivatives you should get pi times this is r squared x minus and then one third times x minus H minus R cubed From x equals zero two H. Plugging in. We get pi times R squared h -1 3rd of yeah are understood- and this is 0 -1/3 of. Sorry there should be r cubed. My mistake I as a third times negative H minus R cubed which is plus one third times h minus r cubed rider bikes at each other and then try to joe get him through the spokes data, lock the bike up, you fucking fall off. That's what and so simplifying. This is pi times R squared h minus one third R cubed. And this is actually plus 1/3 times ar- H cubed. Yeah you could leave in this form or simplify further so mhm slip behind in the bars and then got stuck. If you multiply everything out this will eventually simplify too. One third times kids are spear liss. We factor that one third pi times three R squared H minus R cubed plus ar minus H cubed. And then we get one third pie and after several steps three R h squared minus H cubed which is equal to one third pie H squared Times three AR- H dump every girl that gets her period and disrespect going to come. You gonna try in the period with me? It's