Integrating in Spherical Co-Ordinates.

[Chaz706]

I have the following Integral

$$\int ^1 _0$$ $$\int _0 ^\sqrt{1-x^2}$$ $$\int _0 ^\sqrt{1-x^2-y^2}$$ $$\frac{1}{1+(x^2)+(y^2)+(z^2)} dzdydx$$

(With the limits working properly!)

Converted to spherical Cor-ordinates, I have

$$\int ^\frac{\pi}{2} _0$$ $$\int _0 ^\frac{\pi}{2}$$ $$\int _0 ^1$$ $$\frac{1}{1+\rho} \rho^2 sin(\phi) d\rho dr d\phi$$

I've converted the function, but how would I start integrating?

 

[dextercioby]

Alright.Transform it in spherical coordinates.But u need the limits...

Daniel.

 

[Chaz706]

I have the limits... it's just that I can't get them right on Latex (stupid coding! I'm getting it right, it's just not displaying it that way!)

Hang on...

EDIT: Problem above now has working limits, and my original question as intended.

 

[whozum]

A substitution MIGHT work, but i would probably go for integration by parts. Remember sin(phi) is constant for the first integral.

 

[dextercioby]

$$\frac{r^{2}}{1+r}=r-1+\frac{1}{1+r}$$

is all u need.

Daniel.

 

[arildno]

Your integrand in spherical coordinates should be: $$\frac{\rho^{2}\sin\phi}{1+\rho^{2}}$$

 

[Chaz706]

I've got this thanks. Thanks to a form of integration in the back of my book.