Volume of a SphereDate: 04/21/98 at 14:08:55 From: Ariel Fishman Subject: Calc 2 proof Can you help me prove the formula for the volume of a sphere? I have no clue as to where to start! Your help with previous questions has been greatly appreciated and very helpful. Thank you! --Ariel Fishman Date: 04/21/98 at 17:15:31 From: Doctor Jen Subject: Re: Calc 2 proof Think about a circle, radius a, centre (0,0). The equation of this circle is x^2 + y^2 = a^2, yes? If you've done circles, you'll know this - if not, use Pythagoras' theorem to convince yourself. You get a sphere by taking this circle and rotating it about the x-axis. Once you have your sphere, imagine it chopped up into lots and lots of slices, cuts going vertically. It's like when you slice a tomato. One of these slices, we say, has radius y, and width (delta x). The volume of this element is approximately equal to pi*y^2*(delta x) (the area of the disc, times its thickness. It's approximate, because it's not quite a regular cylinder, being a slice of a sphere - the radius at the top is a bit bigger than the radius at the bottom). The volume of the sphere, then, is the sum of all these elements, added together between x = -a and x = a. (Please try doing a diagram to follow this through; it should make a lot more sense. Well, when you've sliced the tomato, you put the slices together to make the whole again - yes?) Thus, V is approximately SIGMA (pi*y^2*(delta x)). This is still approximate, because we assumed the element was a disc in order to calculate its volume, when in fact its edge was slightly curved, so it wasn't a true cylinder. So we let delta x -> 0. This makes the element become closer to being a cylinder as its width becomes smaller. Now we can say V = int. pi*y^2 dx between -a and a. If you remember, y^2 = a^2 - x^2. So substitute this into the integral, to get V = int. pi*(a^2-x^2) dx between -a and a. Okay from there? Integrate, apply the limits, and find that V = (4*pi*a^3)/3 Enjoy! -Doctor Jen, The Math Forum Check out our web site! http://mathforum.org/dr.math/ Date: 04/21/98 at 17:43:46 From: Doctor Anthony Subject: Re: Calc 2 proof There are many different methods, all using integral calculus. If you consider a circle x+2 + y^2 = a^2 and rotate this about the x axis you generate a sphere. Using the usual formula for a volume of revolution: V = INT(-a to a)[pi*y^2*dx] = INT[pi*(a^2-x^2)dx] = pi*[a^2x - x^3/3] evaluated from -a to a = pi[a^3 - a^3/3 -(-a^3 + a^3/3)] = pi[2a^3/3 + 2a^3/3] = 4pi*a^3/3 -Doctor Anthony, The Math Forum Check out our web site! http://mathforum.org/dr.math/ For more about sphere formulas, see the Dr. Math FAQ: http://mathforum.org/dr.math/faq/formulas/faq.sphere.html |
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