# Different results for NIntegrate for the same function using cartesian and polar coordinates

This is probably a straightforward question. I have two functions f[x_,y_,z_] and g[r_, theta_, z_]

where:

g[r_, theta_, z_] returns f[ r Cos[theta], r Sin[theta], z].

For the same points in cartesian and polar coordinates; f[x,y,z] and g[r, theta,z] return the same values. For example:

f[1.78,1.78,0.5] returns 10.93 g[2.517, Pi/4, 0.5] returns 10.93

However NIntegrate on f[x,y,z] and g[r, theta,z] returns different values.

NIntegrate[f[x,y,z] Boole[x^2 + y^2<=1.78], {x,0,1.78},{y,0,1.78},{z,0.4,1.4}]


returns 25.34

NIntegrate[g[r,theta,z], {r,0.1.78},{theta,0, Pi/2},{z,0.4,1.4}]


returns 195.4

My first thought is that each call is evaluating a different set or number of points over the same region.

Are there any diagnostic functions that may be used to look at the points evaluated by NIntegrate?

For example does NIntegrate use the same data generated by:

Table[{r, Theta, z}, {r,0.1.78},{theta,0, Pi/2},{z,0.4,1.4}]


g[r_, theta_, z_] := f[r Cos[theta], r Sin[theta], z]