Vol of sphere =4/3(pi)r

^{3}

to derive this eqn using polar coordinates

Vs=int r

^{2}drd@sin&d&

limits are r=0,r; @=0,2pi; &=0,pi

were does the d& come from?

i.e. there is no dimension d&

i.e. in deriving eqn for Area of circle, A

_{circle}=int rdrd@

rd@ and dr are dimensions of a (very small) area

so in A

_{circle}=int rdrd@ everything in the integrand is accounted for.

but, in the 3rd dimension, rsin& is the depth of this area, making a (very small) volume.

so V

_{sphere}=r2drd@sin&___

there is no d& in the integrand.

I am not able to supply a picture or diagram.

but, draw your own consisting of two pictures,

a full circle above, and the bottom half of a circle below, both in polar coordinates.

circle uses angle @, half circle uses angle &

then the infinitely small volume has dimensions rd@, dr, and rsin&.

there is no d& dimension.