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Problem 36 Easy Difficulty

(a) Show that the surface area of a zone of a sphere that lies between two parallel planes is $ S = 2 \pi Rh $, where $ R $ is the radius of the sphere and $ h $ is the distance between the planes. (Notice that S depends only on the distance between the planes and not on their location, provided that both planes intersect the sphere.)
(b) Show that the surface area of a zone of a $ cylinder $ with radius $ R $ and height $ h $ is the same as the surface area of the zone of a $ sphere $ in part (a).

Answer

a) Let $a$ be the location of one plane and $a+h$ be the location of the other
plane. Evaluate a surface of revolution integral with those as limits, where
the curve is from a circle of radius $R$ .
b) Let $a$ be the location of one plane and $a+h$ be the location of the other
plane. Evaluate a surface of revolution integral with those as limits, where
the curve is from a line $x=R$ .

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Video Transcript

Okay, So, yeah. Uh, you know the surface. Um, It's here. We'll reduce our So you draw this here. The X Y plane, Maybe something like these more around, right? Yeah. That, Uh yeah, this is the X axis. The y axis in the distance. Here's our on. Then the equation for goddess fears X squared plus y squared. This is what you are. So I want to consider or this is a here. What is the area obtained by cutting to follow planes that intercept this fear and our distance age? So what is the area of this on their lying region between the two planes? Eso The region is gonna look something like this on. Uh, Well, since we can online, those rotated those two planes. So they are. So the reason would be something like that. So we can Yeah, a line. Does. The planes were taken. Sort of. The is gonna be, like, considering work the region between these two plains, but our power to the egg y axis. So distance heroes age. So what? We would have two numbers. Uh, so this distance here you say distance there is B so that a plus B is equal to age. Well, this point here is gonna be since it is central zero this point, he will be on this point Over here is minus safe. So ah, with 15 the area off these would be to rotate this piece of the curve about the X axis so we would rotate this piece of liquor between minus a on me. It would rotate that around the on the exact instead of you. You have our area. So, uh, uh, s O that this area is that so the area is gonna be the area off this vision. This is the same, the same area. Is that so? The area off these, uh, since this is the X y axis theory off this this area is one of the they developed from minus a Have to be. I'm gonna say it could be, and then we'll dysfunction. We're here is, uh, the positive solution for why, over peace. So we have that. Why is he going toe moving these over outside on the square root? Because we only care about this positive part with the squared off. All right, square minus X squared. So this is our function is a VFR wife. Uh huh. Why physical toe ffx is your apple fix So that the interval would be okay, major from minus a upto me off to pie himself. Effects square root off one plus if prime squared Um, yeah, The X s o for dysfunction with prime is equal to well, we have this crude so we have one half, whatever. That's word, uh, thes in terms the Nativity that that is two x minus two x so that minus two x So the two cancel on the the tutu's canceled So air prime square would be X squared My sex squared x squared over over duck over these You know, the square root So are square miles Texas quirk so that these would be square root off on one plus if Brian Square would be the square root one was that Lexus Square over our square minus X squared. And then having these two, this will be the square root off one times that. So making contaminate er our square minus X square over our square minus 60 square. This is our one did not plus next square over our square, minus six square on and he simplifies toe all these x square council. Yeah. Do you have a better on the top half square root of R squared over the square root more off fire squared minus x squared. So this would be that then our function This this So this is integral is gonna be this area It's gonna be ableto to five comes into a from Marseilles to be off Square it off R squared minus X squared times These will square The bar square is just our for our positive are over square root off r squared minus x squared on that the x something very convenient happens is to cancel so that these interval this area will be by central from minus a have to be are on the X So since those are constantly would be to buy our in 12 of the eggs But one of the eggs is just exhale worked out Thio endpoints So goodbye And some well x b So do minus that a minus a would be minus minus saved on Then we have the are there. So to fight them these times are have you obtained 25 are and I think, plus building a plus B on There were all these equals these h the distance, which is the distance between the two planes so that the the area is gonna be that right and sign from sage on the well, Something that does look, a little bit of strangers that well, we have. Ah, sitting there are where you are so separately. We have a ceiling there or what? You Sorry. It is just in there. Seeing the off these ladies here are, um Now we consider the distance surface area of our the resulting these of all cutting again by two planes part of links that are so on this super planes and then the distance between these true these distances age so well, since this is a particular region becomes unfold. These, uh, this piece of the cylinder, my level is gonna be these. So you have what she would have something like now is on the dot on top from the bottom. So that way unfolded eyes, bad vision. And then, with this region here, that is the perimeter very simple to to buy. Times are reduced under this distance. Here is age, so that this area he said you're here. Um, it is also this number because here is that distance times that distance since you deserve rectangle. So it's kind of age times to pipe. I'm sorry. Which is the same area as if cut us here by two planes by two parallel planes that are suffering a distance age. So both areas are are you? Well, this is the same or two is the order your numbers? It is the same.