Calculating area of sphere within particular angular range

Hello everyone,
I have a confusion because of different conventions used by mathematician and physicists for defining spherical coordinates. I have measured Far field data and now I am trying to calculate Total radiated power for it. I am using this formula for first calcuting power through small area

Here you can see the the range of theta of is from 0 to 360 and range of phi is 0 to 180. But in measured data I have ranges for theta from 0 to 360 and phi 0 to 180. But using defination for azimuth and elevation from this site

http://mathworld.wolfram.com/SphericalCoordinates.html

I understand theta is my azimuth and phi is my elevation. So will that means the formual I have pasted above the sin(theta)d(theta)d(phi) should be change to sin(phi)d(phi)d(theta) or it will be same.

With the angles as in the site provided, then, the area of a sphere is as follows:

Area of Sphere =∫^azimuth ∫^elevation R2 sin(elevation) d(elevation) · d(azimith)

All integrals lower limit is 0.

If you change azimuth with 2Π and elevation with Π, you will get 4ΠR2.

Call them however you want... variables are just.. variables.