Solid of revolution volume
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Disc method: function rotated about x-axis
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Disc method (rotating f(x) about x axis)
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Volume of a sphere
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Disc method with outer and inner function boundaries
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Shell method to rotate around y-axis
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Disk method: rotating x=f(y) around the y-axis
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Shell method around a non-axis line
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Shell method around a non-axis line 2
Volume of a sphere Figuring out the equation for the volume of a sphere.
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- Welcome back.
- I don't know what I was thinking.
- Sometimes my brain malfunctions.
- But just going back to that problem we were doing, actually
- I think we should do it.
- I'm a little schizophrenic today.
- So let's figure out the equation for the
- volume of a sphere.
- So what's the equation?
- It's x squared plus y squared is equal to r squared.
- And let's just write y as a function of x, just so
- we can do it the way we did that last problem.
- So you get y squared is equal to r squared minus x squared.
- y is equal to the square root of r squared minus x squared.
- And let's draw it.
- So if this is my y-axis, this is my x-axis, and the
- equation-- draw it straight-- that's my x-axis, and then I
- actually have a circle tool, let me see if I can use it
- effectively-- well, close enough.
- There you go.
- I think you get the point.
- But anyway.
- This is just going to be the upper half of the circle.
- Actually, I should probably just undo that circle tool
- and try to draw it by hand.
- So y equals the square root of r squared minus x squared.
- That's just going to be the upper half of the circle.
- So it will be the positive x quadrant-- and then actually,
- I should have drawn the whole hemisphere.
- But anyway.
- Actually, let me do that, because I think it'll
- make-- edit, undo, let me clear all of this out.
- Sorry for wasting your time, but I think it'll be effective.
- OK.
- So let me redraw.
- So this, that's the y-axis, that's my x-axis, and then
- this-- the square root is, since it's a function, it can
- only have one value, so we assume it's defined as the
- positive square root.
- So if we were to graph that, it would look like this.
- Something like that, where this would be minus r and that's r.
- So if we want to find the volume of a sphere with radius
- r, we just have to rotate this function around the x-axis.
- This is the x-axis, that's the y-axis.
- So let's see what we can do.
- Let's just visualize the disks again.
- So let me make a disk.
- So let's say that that's the side of one of the disks again,
- and as we know, the depth of the disk is just
- going to be dx.
- That's how wide that disk is, dx.
- And its radius at any point is f of x, and in this case, it's
- y is equal to square root of r squared minus x squared.
- So what's the surface area of each disk?
- What's this?
- The surface area of each of the disks.
- I hope you know what I'm saying.
- So area is equal to pi r squared, the radius at any
- point is equal to this, radius is equal to y which is equal to
- square root-- and remember, this is not this r.
- This is the radius of this disk.
- I know it might be a little confusing.
- y is equal to the square root of r squared minus x squared.
- So the area is going to equal pi times this squared.
- So if you square this quantity, you just get rid of the
- square root sign, right?
- So pi r squared minus x squared, and that's the
- area, and so what's the volume of that disk?
- Well just like we've done in every video up to this point,
- the volume of that disk is just that, so the volume of that
- disk is just this pi r squared minus x squared times dx.
- And so if we want to figure out the volume of all these disks,
- I have a disk here, a disk here, going around and around
- and around and around and around and they get smaller and
- smaller until we have a sphere.
- We just take the integral, the upper bound is positive r, the
- lower bound is minus r, and we take the integral of
- this expression.
- pi-- let me distribute it, because that's going to make it
- easier-- pi r squared, which is just a constant term, minus pi
- x squared, all of that dx.
- So what's the antiderivative of that expression?
- The antiderivative within the parentheses.
- Well, this is just a constant term.
- pi r squared, that's just a number, because we're just
- taking the integral with respect to x.
- So the antiderivative of pi r squared is just pi r squared x,
- the derivative of pi r squared x is just pi r squared, minus--
- and we did this in the last video.
- Actually, well now, it's the antiderivative x squared, which
- is x to the third over 3, and the pi is just a constant, so
- pi x to the third over 3, and we're going to evaluate
- that at r and minus r.
- Let me erase some stuff, looks like I'm running out of space.
- Hopefully all of that you know by now.
- OK, back to the pen tool.
- OK.
- So let's evaluate it at r.
- So this is pi r squared, and then for x, we'll substitute
- the positive r times r minus pi x cubed, but now we have this r
- here, so r cubed over 3 minus pi r squared, and then we have
- a minus r here, because we're evaluating the antiderivative
- at minus r, times minus r, minus pi minus r cubed.
- So what's minus r cubed?
- It's r cubed, but we'll keep the minus sign.
- r cubed, and at that minus sign, let's just make
- that-- that'll turn that into a plus-- over 3.
- Let's see if we can clean this up a little bit.
- So that first term is pi r cubed, r squared times r, minus
- essentially 1/3pi r cubed.
- And then, what is this?
- This is pi r cubed, but then we have a minus sign up.
- This is minus pi r cubed, and then we have a minus sign up
- here, so this becomes plus pi r cubed, and then minus-- because
- we have a plus here and a minus out here, so distribute it--
- so minus 1/3pi r cubed.
- And let's see, what do we have?
- We have essentially 1.
- If we just distribute out the pi r cubes, we have pi r cubed
- times 1 minus 1/3 plus 1 minus 1/3.
- Well that's 2 minus 2/3, or another way, let's see, is 2
- minus 2/3-- this is turning into a fractions problem-- and
- what's-- well, that's 6/3 minus 2/3, it equals 4/3.
- So this is equal to 4/3.
- So the volume of the sphere is equal to 4/3 pi r cubed, which
- is the equation for the volume of a sphere.
- And actually, now that I realize, it did take
- me eight and a half minutes, so I am glad.
- My first intuition is always correct, I am glad I did
- this in a separate video.
- But that should be pretty interesting to you.
- And it makes a lot of sense.
- It's going to be a cube of the radius, pi is involved.
- The 4/3 is interesting, just in terms of how it relates
- to everything else.
- Area is pi r squared, and then all of a sudden you get a 4/3
- here, so it is something for you to think about.
- Anyway, hopefully you found that fun.
- I'll see you in the next video.
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At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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