Compute the following integral in spherical coordinates system:
- $$\int_{0}^{1}\int_{0}^{\sqrt{1-y^{2}}}\int_{0}^{\sqrt{1-x^{2}-y^{2}}}\left(\frac{1}{\sqrt{z}}\right)dzdxdy$$
I think the first triple integral is a sphere in the first octant, so the integral is equivalent to : $$\int_{0}^{\frac{\pi}{2}}\int_{0}^{\frac{\pi}{2}}\int_{0}^{1}\rho^{2}\sin\varphi\frac{1}{\sqrt{\rho\cos\varphi}}d\rho d\theta d\varphi=1.2566370\color{red}{4825}$$
But $$\int_{0}^{1}\int_{0}^{\sqrt{1-y^{2}}}\int_{0}^{\sqrt{1-x^{2}-y^{2}}}\left(\frac{1}{\sqrt{z}}\right)dzdxdy=1.2566370\color{red}{5686}$$
So why the answers are not the same?
Concerning the first integral, the value is $\frac{2\pi}5$, both when you use Cartesian coordinates and spherical coordinates. So, that difference that you get is just due to a rounding error.