Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

I Volume of sphere using integration??

  1. Aug 7, 2016 #1
    Is it possible to find the volume of a sphere(i know the formula) using definite integration ???? And if possible how to proceed ??
    Thanks in advance
     
  2. jcsd
  3. Aug 7, 2016 #2

    BvU

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Hello Prasun-rick, :welcome:

    Wat is the formula you know ? And what do you know of integration ? Does ##\iiint dV ## mean anything to you ?
     
  4. Aug 7, 2016 #3
    Well I only know the geometrical formula of volume of the sphere (i.e frac{4/3}pi*r^3)..and I only happen to know integration in one dimension ..though your integral didn't make any sense !! What I will have to do to understand that??
     
  5. Aug 8, 2016 #4

    BvU

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    In that case (working the three dimensional integral into a one-dimensional integral): do you know the volume of a disk that is obtained by revolving a rectangle around one of its sides ? (disk thickness a and radius b when revolving around the x-axis in the figure below)
    upload_2016-8-8_10-1-4.png
     
  6. Aug 8, 2016 #5
    No sir I am afraid I don't know that !! Maybe the derivation is above my scope !! Btw thanks for your valuable comments !! I will get back when I have done the volume Integral of the disk to you for further discussion .
     
  7. Aug 8, 2016 #6

    BvU

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    View attachment 104413
    Maybe I confused you. The result of this revolution is a disk like this and that volume is relatively easy to express in a and b ...
     
  8. Aug 8, 2016 #7
    Can you just pose the integral equation of the volume of the figure you posted !
     
  9. Aug 8, 2016 #8

    BvU

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    It's not an integral. It's a disk. I lost the picture, here it is again, with the question: what is the volume, expressed in a and b ?:

    upload_2016-8-8_14-21-12.png
     
  10. Aug 8, 2016 #9
    Will it be pi*b^2*a??
     
  11. Aug 8, 2016 #10

    BvU

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    It certainly is ! Now we are going to slice a sphere with radius r into disks of thickness ##dx##. What will be the volume of the disk at ##x## ?

    upload_2016-8-8_14-34-30.png
     
  12. Aug 8, 2016 #11

    BvU

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Maybe this is better :
    upload_2016-8-8_14-37-37.png
     
  13. Aug 8, 2016 #12
  14. Aug 8, 2016 #13

    BvU

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Perfect. Next step: we are going to add up all these disks from -r to +r and if we take the limit for ##dx\downarrow 0## we get an integral. Can you write down that integral ? (it's an easy question, because you have almost all of it already...)
     
  15. Aug 8, 2016 #14
    yeah maybe ∫pi*(r^2-x^2)dx with integral limits from -r to r ??
     
  16. Aug 8, 2016 #15

    BvU

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Bingo
     
  17. Aug 8, 2016 #16
    Thanks :smile:
     
  18. Aug 9, 2016 #17
    But what is that
    Code (Javascript):
    /iiint dV
    and how to solve it ??
     
  19. Aug 9, 2016 #18

    BvU

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    It means you want to sum up infinitesimal volume elements ##dV##. A volume like a sphere has three dimensions, so three integrations are necessary. In cartesian coordinates you get the volume of a cube at the origin with size ##a## from $$
    \int\limits_0^a \int\limits_0^a\int\limits_0^a dxdydz = a^3$$For a sphere the limits are unwieldy in cartesian, but comfortable in spherical coordinates. A volume element is ##r^2drd\phi d\theta## (##\theta## is azimuthal) so you get $$
    \int\limits_0^{2\pi} \int\limits_0^\pi \int\limits_0^R \; r^2 \sin\theta \;dr \;d\theta \; d \phi = \\
    2\pi \int\limits_0^\pi \int\limits_0^R \; r^2 \sin\theta \;dr \; d\theta =
    4\pi \int\limits_0^R \; r^2 \; dr = \ ...
    $$

    (There are alternative notations, like ##\ \int\limits_{\rm Volume} d^3 V\ ##)

    [edit] corrected order of ##d## in expressions but I think I still have it wrong. Need to check if we work outside in or inside out o:)
     
    Last edited: Aug 9, 2016
  20. Aug 9, 2016 #19
    Is it okay for me to learn triple integral now?? And if yes then where to start??
     
  21. Aug 9, 2016 #20
    And please teach me how to give such prominent integral sign like the ones you are typing
    !?
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted