Maybe just use
or do you have to use spherical co-ordinates?
i'm having little problem with this .... i need to find volume of figure
using spherical coordinates ...
now i realize that this is just a sphere, actually half of the sphere with center in (1,0,0) and radius of 1 ... that is not the problem... and it's volume should be as half of volume of same sphere in center (0,0,0) ...
but i'm having problem how to figure out limits for variables ...
i know that
but how does them look like that if figure is shifted from (0,0,0) ? or is it something like :
i need to use something like
and J should be the same ? ? ? ?
any help will be very much appreciated
i know that for let's say ellipse, if i need to use polar and it's shifted i'll use
but can i apply same thing here ?
the issue here for that sphere is how to find "r", limit's for "r"
i try to solve that like this but get nowhere
so if apply spherical coordinates
so r will go from to ???
and from to
and from to
where do i go wrong ?
that will mean that
now because i know that r is always it will mean that ?
so from that i find limits ? when the sin is positive ? sin is positive in first and second quadrant ? correct ? and i know natural limits of the and that so it will mean that i have limits
I agree with your result, but I think that in the process of getting there, you confused the polar angle with the azimuthal angle. (This is something the mathematicians and physicists wrangle about, and concerning which they use different notation.) You can always tell which one is which by the angle that appears in the z component. That's always the polar angle, as far as I know. Its natural limits are The azimuthal angle corresponds to the angle used in cylindrical coordinates and polar coordinates, and is measured positive in the counter-clockwise direction from the positive x axis, as per usual. The azimuthal angle always inhabits
you probably know that, but just to put it here so, one that reads those equations will know why is it two "different" spherical coordinates that you wrote there (it's the same but... )