# Integral over an area of spacetime [closed]

Is it possible to evaluate this integral in spacetime?

$$\int_{\Sigma} \frac{dydz}{[a_{o}(y^{2}+z^{2})+2f_{o}y+2g_{o}z+c_{o}]^{2}}$$ where $$a_{o}c_{o}-f_{o}^{2}-g_{0}^{2}=\frac{1}{4}.$$

If it is possible, can someone shed some light on how can this be done?

• If this is over spacetime, should the measure be $\sqrt{-g}d^4x$? – Gradient137 Nov 10 '18 at 7:02
• Have you tried shifting $y,\,z$ by the obvious constants? – J.G. Nov 10 '18 at 7:03

I'll offer a hint, expanding on my comment. Define $$y':=y+\frac{f_0}{a_0},\,z':=z+\frac{g_0}{a_0}$$. The quadratic becomes $$a_0(y'^2+z'^2)+c_0-\frac{f_0^2+g_0^2}{a_0}=a_0\bigg(y'^2+z'^2+\tfrac{1}{(2a_0)^2}\bigg).$$So now you just need to evaluate$$\frac{1}{a_0^2}\int_\Sigma\frac{dy'dz'}{\bigg(y'^2+z'^2+\tfrac{1}{(2a_0)^2}\bigg)^2}\cdot$$The substitution $$u=y'^2+z'^2$$ might be worthwhile.