\[x = r\cos(\theta)\hspace{1cm}y = r\sin(\theta)\hspace{1cm}z = z\] \[0\leq\,r\leq\sqrt{z}\hspace{1cm}0\leq\,\theta\leq2\pi\hspace{1cm}0\leq\,z\leq1\]

A solid object exists within the standard \((x,y,z)\) coordinate system. Its geometry is parametrized as shown above.

The object has a mass density \(\large {\rho = e^{z^{2}}}\), where \(e\) is Euler's number. The objects mass can be expressed as:

\[{\dfrac{\pi}{A} (e - B)},\]

where \(A\) and \(B\) are integers, determine \(A+B\).

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