Polar coordinate integration in different planes?

[theBEAST]

I know that when you are integrated over dA in the xy plane, for your polar coordinates, x = rcosθ and y = rsinθ. However what about in the xz and yz plane?

I noticed in one of the textbook problems, where the integration is over an area in the xz plane, x = rcosθ and z = rsinθ. How did the solution know to use that?

 

[clamtrox]

What do you mean? Why would xy and xz-planes be any different? You can get xz-plane from xy-plane by just a simple rotation

 

[theBEAST]

What do you mean? Why would xy and xz-planes be any different? You can get xz-plane from xy-plane by just a simple rotation

If you integrate over some polar area in the xy plane you find that x = rcosθ and y = rsinθ. but let's say instead of integrating over the xy plane, we integrate over the yz plane then what are y and z in terms of r and θ?

 

[clamtrox]

If you integrate over some polar area in the xy plane you find that x = rcosθ and y = rsinθ. but let's say instead of integrating over the xy plane, we integrate over the yz plane then what are y and z in terms of r and θ?

Well, how do you define r and θ? I'm guessing r is the distance, so r=√(x²+z²) and θ is some angle, but what exactly is it?

 

[SteamKing]

r, theta polar coordinates are for the plane only. If you have a three-dimensional problem to analyze, then spherical coordinates (r, theta, phi) would be called for.

The conversion of polar coordinates (r, theta) to Cartesian (x, y), where x = r cos theta, y = r sin theta, is just simple trigonometry.