1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Volume of a sphere (solid of revolution)

  1. Jan 21, 2009 #1
    EDIT: [so sorry guys i thought I had clicked h/w c/w calculus section, if any admin could move this thread there that would be really appreciated, sorry for posting in the wrong sub-forum]

    Hi guys, I am having an issue with the volume of a sphere and calculating it. I am using the method where the solid, sphere in this case, is composed of many cylinders, in my case along the x-axis where the center of the sphere is at the origin.

    So initially I attempted this by considering each cylinder was of volume:[itex]\pi y^2 \delta x [/itex]
    then the formula of the circle in the xy plane of the sphere is [itex]x^2 + y^2 = r^2[/itex] which yields [itex]y^2 = {r^2 - x^2} [/itex] which means the volume of each cylinder becomes [itex]\pi (r^2 - x^2) \delta x[/itex]

    I will limit my range to [0,r] as it is symmetrical about the yz plane. So integrating accros the interval [0,r] we get :

    [tex]V = \pi \int_{0}^{r} (r^2 - x^2) \delta x [/tex]
    [tex] = \pi ({r^2}x - \frac{x^3}{3})\bigg{|}_0^r [/tex]
    [tex] = \pi [({r^3} - \frac{r^3}{3})-({r^3}(0) - \frac{0^3}{3})] [/tex]
    [tex] = \pi \frac{2r^3}{3}[/tex]

    then x2 to get the volume of a sphere to be [itex]{4\pi r^3}{/}{3}[/itex]

    now thats all well and good, but I also wanted to see if another method would work, instead of using x as our independent use theta. Now I learnt that if you wanted to calculate the surface are of a sphere you can use theta as it makes calculations easier. So using the idea that we can calculate the surface are of a sphere modeling it as composed of many hollow and cylinders with no base or head, each has height [itex]r\delta \theta[/itex]. So each cylindrical surface becomes :

    [tex]i. \ A = 2\pi y r \delta \theta[/tex]
    [tex]ii. \ y=rsin\theta[/tex]
    [tex]iii. \ A = 2r\pi rsin\theta \delta \theta[/tex]

    So thats fine, that idea can be taken as can successfully yield the surface area of a sphere successfully. Now I figured if that method can be employed for a surface, it must be adaptable to find the volume of a sphere. so this time I decided to define each solid cylinder as:

    [tex]iv. \ V = \pi y^2 r \delta \theta[/tex]
    [tex]v. \ V = r\pi (r^2sin^2\theta) \delta \theta [/tex]

    so again I approached this as before, this time integrating across the range [0, [itex]\pi / 2[/itex]] where I again will only find the hemisphere for +ve x and theta is the angle from the x axis. So my integration went like this :

    [tex]V = \pi r^3\int_{0}^{\frac{\pi}{2}} (sin^2\theta) \delta \theta [/tex]
    [tex] = \pi r^3\int_{0}^{\frac{\pi}{2}} \frac{1}{2}(1 - cos2\theta) \delta \theta [/tex]
    [tex] = \frac{\pi r^3}{2}(\theta - \frac{1}{2}sin2\theta)\bigg{|}_{0}^{\frac{\pi}{2}}[/tex]
    [tex] = \frac{\pi r^3}{2}(\frac{\pi}{2}) = \frac{\pi^2 r^3}{4} [/tex]

    then oviously x2, but as is evident it is not the right answer, I really cant see what I have done wrong, I pretty sure none of my actually mathematics is wrong, which means the only part that could be wrong is the [itex]r\delta \theta[/itex] bit and teacher suggested that for some reason it doesn't apply to a solid but does to a shell, but couldn't give a valid reason as to why. I am inclined to agree with him, but again I cant see why. In various texts I have looked though it says things along the lines of for the shell it is tempting to use [itex]\delta x[/itex] but that [itex]r\delta \theta[/itex] is a "better approximation", therefore surely it to is a better approximation for a solid of revolution to.

    Can anyone explain why this is, what is the reason the integration cant be done this way, or am have I simply done my maths wrong, thanks guys :D
  2. jcsd
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted