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You are not logged in. #1 20060204 14:12:52
Volume as a function of height in sphere?I'm stumped. I tried related rates, integrating cylindrical shells, and integrating using disks, all to no avail. Every method I try, leaves me in a total mess. #4 20060204 15:14:44
Re: Volume as a function of height in sphere?w00t! Very interesing. Have you considered integrating in terms of y using the pi r^2 method from y = 0 to y = h and adding that to the other half othe sphere? (unless the sphere is less then half full, but again a similar method might be doable.) A logarithm is just a misspelled algorithm. #5 20060204 15:23:57
Re: Volume as a function of height in sphere?I never thought of mixing methods. But the disk method: ∫πf(y)²dy rotates about the x axis while the shell method ∫2πxf(x)dx rotates about the y axis. The two I would think are incompatible. #7 20060204 15:41:09
Re: Volume as a function of height in sphere?
forgive me if I'm wrong but isn't the disk method integrated in terms of x when rotating about the x axis? A logarithm is just a misspelled algorithm. #8 20060204 16:02:21
Re: Volume as a function of height in sphere?I don't know why I typed that. What I was trying to say was that the shell method revolves around the opposite axis of the variable while the disk method revolves around the same axis as the variable. Last edited by irspow (20060204 16:05:23) #11 20060204 20:43:39
Re: Volume as a function of height in sphere?If we work it out with the base of the sphere centered at the origin we don't have to worry about negative integrals, and can do it in one piece. If I integrate it for you... Last edited by mikau (20060204 20:53:16) A logarithm is just a misspelled algorithm. #12 20060205 02:16:09
Re: Volume as a function of height in sphere?Nice, mikau, you have earned the rank of GOD in my opinion. I missed the substitution for R when I was going around in circles a few days ago. I kept seeing that x and, instead of substituting, I was banging my head trying to use related rates for x and y. #14 20060205 05:05:44
Re: Volume as a function of height in sphere?And the wonderfull thing about calculus, now that we have the formula, differentiating allows us to calculate the rate at which the height is changing as the volume increases at a given rate and vice versa. Last edited by mikau (20060205 05:14:12) A logarithm is just a misspelled algorithm. #15 20060205 05:45:42
Re: Volume as a function of height in sphere?Yes, the concept of solids of revolution will always be etched in my mind. There was such a WOW factor to them in my mind when I was first introduced to them. I always find if amazing that many of the foundations for this type of computation were derived from individuals working by candlelight in an age which now seems eons ago. The imagination of the human mind is truly incredible. #16 20060205 05:49:20
Re: Volume as a function of height in sphere?Yeah, but whats most odd is no one can come up with anything new like that anymore. How come people were so much smarter a thousand years ago? Last edited by mikau (20060205 05:56:26) A logarithm is just a misspelled algorithm. #17 20060205 05:58:46
Re: Volume as a function of height in sphere?You must remember that people back then had very few alternatives to spending their time compared to what we now live like. 