A triple definite integral from Cartesian coordinates to Spherical coordinates. Help!

Question

I am trying to solve evaluate this triple definite integral. The integral is given below.

$$ \Lambda =\int_0^3\int_0^{\sqrt{9-x^2}}\int_0^{\sqrt{9-x^2-y^2}} \frac{\sqrt{x^2+y^2+z^2}}{1+(x^2+y^2+z^2)^2} \, dz\,dy\,dx $$

I know that this problem has to be done in spherical coordinates, since we have a triple integral, and we have the following relationship :

$$ x^2+y^2+z^2 $$

and I know that this equals to:

$$ \rho=\sqrt{x^2+y^2+z^2} $$

I know that for spherical coordinates my x,y,z variables have the following form:

$$ x = \rho\cos(\theta)\sin(\phi) \\ y = \rho\sin(\theta)\sin(\phi) \\ z = \rho\cos(\phi) $$

Once I make the substitutions my integral expression gets really complicated, and I have no idea how to proceed from there. Could anybody post a step by step solution to this triple definite integral?And also showing how the change of variables are done within the coordinates and the integral. I do believe that using spherical coordinates for this problem is the best way to solve evaluate this triple integral. Thank You!

Answer 1

we need to deal the sphere in the first octant. so $\theta$ varies from $o$ to $\pi/2$, $\rho$ from $0$ to $3$, $\phi$ from $0$ to $\pi/2$. $$ \Lambda =\int_0^3\int_0^{\sqrt{9-x^2}}\int_0^{\sqrt{9-x^2-y^2}} \frac{\sqrt{x^2+y^2+z^2}}{1+(x^2+y^2+z^2)^2} \, dz \, dy \, dx$$

$$=\int_0^{\pi/2}d\theta\int_{0}^{\phi=\pi/2}\sin(\phi) \, d\phi \int_0^{\rho=3} \frac{{\rho^3}}{1+(\rho)^4} \, d\rho \, d\phi $$ Here you can integrate.

Answer 2

You want to integrate a rotationally symmetric function over an eighth of a ball of radius $3$. Partition this ball onionlike in spherical shells of area ${1\over8}\cdot 4\pi r^2$ and thickness $dr$. In this way you obtain $$\Lambda={\pi\over8}\int_0^3{r\over 1+r^4}\>4r^2\>dr={\pi\over8}\>\log(1+r^4)\biggr|_0^3={\pi\over8}\log 82$$ Doing it all the way in terms of spherical coordinates involves computing the Jacobian of the coordinate transformation.