Find error in mathematical proof of electromagnetic fields
What's missing here is the original question. But I guess it is something like


  For  a hollow sphere with radius \$r_0\$ and surface charge \$\rho_s=1000C/m^2\$ located at the origin of the coordinate system, find the electric field at a point \$\vec{x}_0\$ outside the sphere.


In general, you have two issues in your calculation.

First, it is a surface charge, and \$dq\$ does not depend on any \$dr\$, just \$r^2_0\$:

$$ds=r_0^2\sin\theta\,d\theta\,d\phi$$

$$dq=\rho_sr_0^2\sin\theta\,d\theta\,d\phi$$

So you'll only have to integrate over this two angles, too:

$$\iint...r_0^2\sin\theta\,d\theta\,d\phi$$



Second each small charge \$dq\$ located at a position \$dr\$ creates a tiny field at \$\vec{x}_0\$. Direction and strength depend on the distance between both:

$$\vec{R}=\vec{x}_0-\vec{r}$$

Of course, \$r\$ is different for each \$dq\$, and since you have chosen spherical coordinates, you have to express this in spherical coordinates, too:

$$\vec{r}=\begin{pmatrix}r\,\sin \theta \,\cos \varphi \\r\,\sin \theta \,\sin \varphi \\r\,\cos \theta \end{pmatrix}$$

With this, the integral is

$$\vec{E}(\vec{x}_0)=\iint\frac{1}{4\pi\varepsilon_0}\frac{\vec{R}}{|\vec{R}|^3}\,dq$$

$$=\iint\frac{1}{4\pi\varepsilon_0}\frac{\vec{R}}{|\vec{R}|^3}\cdot\rho_sr_0^2\sin\theta\,d\theta\,d\phi=\frac{\rho_sr_0^2}{4\pi\varepsilon_0}\iint\frac{\vec{R}}{|\vec{R}|^3}\sin\theta\,d\theta\,d\phi$$



Finally, should you really calculate the field this way? The integral is a bit nasty... Typically, one uses Gauß law plus geometry to get the well known result

$$\vec{E}(\vec{x}_0)=\frac{q}{4\pi\varepsilon_0}\frac{\vec{x}_0}{|\vec{x}_0|^3}$$

with

$$q=4\pi r_0^2\rho_s$$

EDIT: Lost the \$\sin\theta\$...