Deriving the Volume of a Sphere Using Spherical Coordinates

[sssddd]

I just need a really good derivation of it using spherical coordinates, like the integral limits.

pictures might help

 

[amcavoy]

I just need a really good derivation of it using spherical coordinates, like the integral limits.

pictures might help

$$\iiint\limits_E{\rho}^2\,\sin{\phi}\,d\phi\,d\rho\,d\theta\quad E:\left\{0\leq\phi\leq\pi;\quad 0\leq\rho\leq r;\quad 0\leq\theta\leq 2\pi\right\}$$

 

[stunner5000pt]

http://mathworld.wolfram.com/Sphere.html

 

[sssddd]

actually i was more interested in how you derived the d phi(that other angle thing) part

Like which integrant belongs to which. Mathworld doesn't show too much of that, the math part I get but I would like to know which angle belong to which. Since there are 3 sets of integral limits, then there should 3 of them, so which belongs which accoring to the equation cavoy posted

 

[amcavoy]

actually i was more interested in how you derived the d phi(that other angle thing) part

Like which integrant belongs to which. Mathworld doesn't show too much of that, the math part I get but I would like to know which angle belong to which. Since there are 3 sets of integral limits, then there should 3 of them, so which belongs which accoring to the equation cavoy posted

From cartesian to spherical coordinates:

$$x=\rho\cos{\phi}\cos{\theta}$$

$$y=\rho\cos{\phi}\sin{\theta}$$

$$z=\rho\sin{\phi}$$

...then use the Jacobian to get the equivalent of dV in terms of phi, theta, and rho.