Estimating Radius for Half of Total Luminosity: Luminosity Integral Help

[Logarythmic]

If the total luminosity is given by

$$L_T = \int_0^{\infty} e^{-(r/a)^{1/4}} r^2 dr$$

estimate the radius [tex]r(a)[/itex] corresponding to half of the total luminosity.<br /> <br /> This would be to integrate from zero to r and get a function $r(a,L)$ but this is impossible so anyone got an idea on how to estimate this?[/tex]

 

[quasar987]

Well the primitive of the integrand exists, you knew that right?

The difficult part is solving for $r_{1/2}$ in $$\int_0^{r_{1/2}} e^{-(r/a)^{1/4}} r^2 dr=L_T/2$$. But computers are good with this.