Your prof is correct; you need to substitute spherical coordinates into the equation for the volume. Your limits are wrong because you are not integrating over a sphere of radius 2 centered at the origin.
why does the prof wrote a mistake there and wrote 2cos theta in there
(i didnt typed the question correctly,the original is
i need to calculate the integral on the volume enclosed by z>=0 and the sphere wich is written in the photo.)
correct i am integration on sphere centered (0,0,1)
in spherecal coordinated the third coordinate is the radius and from the photo and you said it youself that the radius is 2.
when i change the coordinated from cartesian to spherecal
i do it from the sketch,so i get radius from 0 till 2
cant see how the shifted center changes the intervals the way it did.
The spherical coordinates are with respect to the Cartesian axes, not the center of the sphere that you're integrating over. Substituting the expressions for the spherical coordinates into gives .
You can shift the Cartesian coordinates to the center of the sphere but then you need to make the corresponding change to the integrand.
You don't use the expressions relating spherical to cartesian coordinates?? That's absurd. This is why you don't understand your professor's comments. Go back to the beginning of this subject and learn it properly this time. There are no shortcuts, believe me.
i have some linking formulas
between spherecal and cartesian
but we dont use them
we just see the sketch and build the integral by it.
can you tell me from the transision formulas
have these intervals?
you dont use these formulas to build the integral
you do it by the sketch
but here i dont know how to get the 2cosine interval
could you show me the transition?