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A cap of a sphere with radius $r$ and height $h$(IMAGES CANNOT COPY)...

Question

A cap of a sphere with radius $r$ and height $h$(IMAGES CANNOT COPY)

A cap of a sphere with radius $r$ and height $h$ (IMAGES CANNOT COPY)



Answers

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

In the question, we have to prove that the hemisphere in closes the largest volume among all this medical gaps, off fixed spectacle or surface area. So for proving these when it do do this question veggies. So, like this, first of all idiot off spill is equal to is that is equal doom to buy a ledge No, r equals is upon Dubai match on we It equals by EJ squad into s a bone to buy edge my one on one tree edge This equal dough is a pawn to it's minus by by three sq no, this equals well, we change for little point. We have to find real estate so he does Age equals is upon to minus private three in tow three squad, this equal do is upon to minus bye Hitch Square. So we have fine in the squad that is equal to basic on for Dubai edges squad equals on dhe. It just quiet because is upon to buy um, it's equals gaze upon to buy it that is equal. Do so we have proved that it equals r. So this response to there is hence the hemisphere in closes the largest while you. Come on. The spat iCal gaps. Oh, fixed surface area surface idea Kiss No, the volume because one upon three into s into Uganda is upon to buy Because we have this age equals broken That is upon to buy on. We have the volume man's when the country surface area in bridge So this is the morning Sid, answer to your problem.

Okay, so eyeballing me is just gonna go from our minus h to r of pi times R squared which is a radius My ex wife Word. Do you? Why thank this gives me hi time r squared times one when it's why cubed over three evaluated at arm and our mind It's h Okay If you simplify this you get high times are eight squared wise one over three cubes We know that our is equal are big are people too Why times r squared for spit squared over to H squared minus one over three inch cube If you simplify those two yet quiet h over six times three r squared plus squared.

Okay, so here we have a sphere with a cop taking off. So we are going to view it from the side, actually, like this where our circle, um, is being stopped at our minus h here. So instead of going us a full circle will stop short here and revolving this around the X axis. Like so. Okay, so our area is going to be pi r squared. Uh, where are here? Is going to be the height. Why? Like so. Okay, so, uh, area is going to be pi times y squared. But hear why squared, uh, is equal to r squared minus X square. Like so. Okay, so our value it is going to be, uh, the integral of pie. And then our bounds are from this is negative are, um and then this is our minus h. So we're going from negative are two are minus h. Uh um r squared minus X squared. And he acts like so, So taking the integral. We have pie times R squared X minus X cube over three, and we're taking it from negative Are to our minus age to simplify things. I'm gonna factor out a 1/3 someone there. We write it like this. Three R squared X minus X cube and again from negative are two are minus H. So plugging all of the r minus agent ours in for N h is Okay, so we have three r squared R minus each, minus R minus, H Q minus. Okay, Um, minus negative. Three are cute, uh, and then, huh minus, uh, our cube like So. Okay, so let's, uh, simplify that even further. We get pie over three. Um, here we have three are cube minus three R squared, age minus are cute plus three r squared H minus three R h squared plus. Yeah. Plus, uh, H cube. And then also plus three r cubed, uh, three are cube. And then, um, minus r cubed. That's right. Plus are acute as well. Here. Okay. Oh. Oh, wait. Um, sorry. This one should be This here. Should be, uh, oops. This here should be minus our cube. Okay, So combining all our like terms, So first we have, um, the are cubes. So we have three r cubed minus our cube. Um, So this becomes four are cube, and then we have our r squared h, those will just cancel out some. Next we have our H squared so minus three R h squared and then our last term is plus H Q. Okay, And then, if you want, you can simplify this further into pie over three times H plus R and H minus two are square. Either these are acceptable answers that we are done.

So for this problem, we want to you find the volume O sphere that's going to have a cap, remember? So essentially, we are going to be able to think about this as if we're rotating this lower half or lower portion of the circle and about our A Y axis around here. And so essentially, all we need to do is start by finding an equation for our circle. So if we place it here on the origin, the equation for our circle is just going to be expert. Plus y squared is equal to R squared. So whatever our radius is, and we know that we are going to be integrating um Lake said about the Y axis, so are lower limit for our Y. Value is just going to be if our radios with straight down. So that's negative art. And our upper bound is this armada's H so we could go and start setting up our interval. So this will be the interval from negative are up through our minus inch. Since we're rotating about the Y axis, we're going to go ahead and want to use our slicing method. So our volume is going to be pi times our radius which, like I said, it's going to need to be in terms of why. So if we isolate this to get X all by itself, we're going to end up with X is equal to the square root of R squared minus y squared So we can go and substitute this in for f of why into our equation square root of r squared minus y squared, squared Since this is already ist an d wife and from here, all we need to do is integrate like normal. So first, we will go ahead and rewrite our bounds of integration and then, uh, square this entire product So we can basically just cancel out our square root sign and distribute our pie. So our friend of the integral of pi r squared minus pi y squared, Do I and we know how to integrate our first time since it's just a constant. All we need to do is make this the coefficient of a single wide term and then minus. We're going to use power rule for our second expression here. So we know that we're going to keep our coefficient the same. So those this will be pie and we're going to multiply this by the reciprocal of our expert. So since our excellent is too, we're going to multiply it by the reciprocal to you and add one under that says gonna be pi over three times. Why? To that exponents plus one power. So why kids and this is evaluated from negative are through our minds, H. And we know that we are going to be able to solve this using the fundamental theory of calculus. So first we'll go ahead and plug in our minus H for why? And we do that we it pi r squared times, Armanis age. My tie over three times are minus h cute. And then we're going to subtract. I would have the results from playing and negative are here. And I should have made these of outside brackets square in my first term here so anyway, won't reply again. Negative are here. So this will be pi r squared times negative art minus pi over three times negative r cubed. And all we need to do is that before this, um and once we combine all of our like terms and factor all the way, we're going to find that the volume of a sphere is equal to pi over three times H plus our times H minus two r squared and this is in cubic units.


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