Deriving Sphere Volume using Spherical Coordinates: Why 0°-360°?

[MHD93]

I wanted to derive the volume of a sphere using triple integration with spherical coordinates, but instead of taking the limits of θ as (0° ≤ θ ≤ 180°), I chose to take (0° ≤ θ ≤ 360°), and therefore, for φ as (0° ≤ φ < 180°),

Now of course the integral of sin(θ) from 0° to 360° is zero, and the derivation failed..

Shouldn't it be the same result, why.. ?
Thank you

 

[tiny-tim]

I wanted to derive the volume of a sphere using triple integration with spherical coordinates, but instead of taking the limits of θ as (0° ≤ θ ≤ 180°), I chose to take (0° ≤ θ ≤ 360°), and therefore, for φ as (0° ≤ φ < 180°),

Now of course the integral of sin(θ) from 0° to 360° is zero, and the derivation failed..

Shouldn't it be the same result, why.. ?
Thank you

Hi Mohammad!

We usually write the volume element as sinθdrdθdφ because we use only 0 ≤ θ ≤ π …

but the element is really |sinθdrdθdφ| …

if you use that, your integral from 0 ≤ θ ≤ 2π will work