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I'm working on deriving the volume of a sphere by taking circular slices of it with volume V = A*dx. However I'm having trouble, because volumes have three dimensions, and the example methods I've seen do not use three variables:
http://www.math.hmc.edu/calculus/tutorials/volume/
It seems to me that A = pi*r^2, and r^2 = y^2 + z^2. So, I should be integrating from -R to R
pi*(y^2 + z^2) dx
Where R is the radius of the sphere and r is the radius of the circular slice.
I'm getting stuck here. Help!
EDIT - The next step seems to be to state that r = y, which is true where z = 0, but this bothers me...why can this step be assumed?
Originally posted by AThousandYoungWell, if you're adding up slices of a sphere in the x-direction you don't need to worry about any differences in the y- and z-directions, because it's symmetrical. You can simplify things by just looking at a circle on the x-y plane.
I'm working on deriving the volume of a sphere by taking circular slices of it with volume V = A*dx. However I'm having trouble, because volumes have three dimensions, and the example methods I've seen do not use three variables:
http://www.math.hmc.edu/calculus/tutorials/volume/
It seems to me that A = pi*r^2, and r^2 = y^2 + z^2. So, I shoul ...[text shortened]... e that r = y, which is true where z = 0, but this bothers me...why can this step be assumed?
The radius of each slice is simply the distance from the x-axis (y=0) to the y-coordinate of the surface of the sphere (y=SQRT(R^2 - x^2)). Therefore the volume of each slice is:
dV = pi*r^2 dx = pi*(R^2 - x^2) dx
If you integrate this expression from -R to R, you should end up with the formula for a sphere (4/3)pi*R^3.
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Originally posted by AThousandYoungSee...you're summing discs of infinetisemal width dx, area pi*y^2...now the value of y is dependent upon x, (the value of z is irrelevent btw since we are dealing wth circles 🙂 )
I'm working on deriving the volume of a sphere by taking circular slices of it with volume V = A*dx. However I'm having trouble, because volumes have three dimensions, and the example methods I've seen do not use three variables:
http://www.math.hmc.edu/calculus/tutorials/volume/
It seems to me that A = pi*r^2, and r^2 = y^2 + z^2. So, I shoul e that r = y, which is true where z = 0, but this bothers me...why can this step be assumed?
So can you find the relationship between y and x and plug this in for y^2?
Finally your limits are fine
*edit PBE6 got here whilst I typed this 🙂
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Originally posted by PBE6If you look at a circle in the x-y plane then you should integrate with respect to z! The volume of the slices will be pi*r^2, with r a function of x and y, and your infinitisimal value will be dz. Right?
Well, if you're adding up slices of a sphere in the x-direction you don't need to worry about any differences in the y- and z-directions, because it's symmetrical. You can simplify things by just looking at a circle on the x-y plane.
The radius of each slice is simply the distance from the x-axis (y=0) to the y-coordinate of the surface of the sphere (y=SQR ...[text shortened]... te this expression from -R to R, you should end up with the formula for a sphere (4/3)pi*R^3.
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Originally posted by AThousandYoungnope..it will be dx, forget about z...it's value takes whatever value y takes since we are dealing with circles. you wish to evaluate the integral from -R to R of pi*y^2. (or if you are really really bothered about what happened to that pesky z, your area is pi*y*z where z = y)
If you look at a circle in the x-y plane then you should integrate with respect to z! The volume of the slices will be pi*r^2, with r a function of x and y, and your infinitisimal value will be dz. Right?
It would be handy if you could express y in terms of x, again...what is this relationship?
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Originally posted by AgergI think I got it. The x^2 + y^2 is actually not from the equation of the circle slice!
nope..it will be dx, forget about z...it's value takes whatever value y takes since we are dealing with circles. you wish to evaluate the integral from -R to R of pi*y^2. (or if you are really really bothered about that pesky z, your area is pi*y*z where z = y)
It would be handy if you could express y in terms of x, again...what is this relationship?
You make a triangle with the radius of the sphere R, making sure to keep the radius you're going to use in the xy plane. This gives you a triangle with R being the hypotenuse and two legs - x and y. You use the Pythagorean Theorem to get the R^2 = x^2 + y^2, instead of the equation of the circle slice, which is r^2 = y^2 + z^2! r, the radius of the circle slice, is the same length as this y, as can be seen if r is oriented in the y direction.
Rearranged, we have
y = (R^2 - x^2)^(1/2)
Which is simply a function of x, and then we integrate from -R to R
pi*r^2 dx
pi*y^2 dx
pi*(R^2 - x^2) dx
Originally posted by AThousandYoungrock on! 🙂
I think I got it. The x^2 + y^2 is actually not from the equation of the circle slice!
You make a triangle with the radius of the sphere R, making sure to keep the radius you're going to use in the xy plane. This gives you a triangle with R being the hypotenuse and two legs - x and y. You use the Pythagorean Theorem to get the R^2 = x^2 + y^2, i ...[text shortened]... circle slice, is the same length as this y, as can be seen if r is oriented in the y direction.
Originally posted by AThousandYoungIf you do that you get a cylinder, not a sphere. Ooh, the math of circles on a Saturday night...
If you look at a circle in the x-y plane then you should integrate with respect to z! The volume of the slices will be pi*r^2, with r a function of x and y, and your infinitisimal value will be dz. Right?
Originally posted by sasquatch672No, the radius of the circles varies with respect to z. You will get a circle if you write down the correct relation between z and the radius of the circle in the xy plane.
If you do that you get a cylinder, not a sphere. Ooh, the math of circles on a Saturday night...