derive eqn for vol of sphere using polar coordinates.
Vol of sphere =4/3(pi)r3
to derive this eqn using polar coordinates
Vs=int r2drd@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, Acircle=int rdrd@
rd@ and dr are dimensions of a (very small) area
so in Acircle=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 Vsphere=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.
Vol of sphere =4/3(pi)r3
to derive this eqn using polar coordinates
Vs=int r2drd@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, Acircle=int rdrd@
rd@ and dr are dimensions of a (very small) area
so in Acircle=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 Vsphere=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.