A Surface Integral - 2

Problem solved in spherical coordinates. Simplifications are tedious.

Problem treated in spherical coordinates:

$x = R \sin{\theta}\cos{\phi}$ $y = R \sin{\theta}\sin{\phi}$ $z = R\cos{\theta}$

$\vec{r} =\left[\begin{matrix} x \\ y \\ z \end{matrix}\right] \ ; \ \vec{r}_o =\left[\begin{matrix} 0 \\ 1 \\ 2 \end{matrix}\right]$

The expression:

$I=\left(\vec{r} - \vec{r}_o\right)^T\left(\vec{r} - \vec{r}_o\right) = 25\,{\cos\left(\phi\right)}^2\,{\sin\left(\theta\right)}^2+{\left(5\,\sin\left(\phi\right)\,\sin\left(\theta\right)-1\right)}^2+{\left(5\,\cos\left(\theta\right)-2\right)}^2$

This is a scalar function. The surface integral is:

$\int_{0}^{2 \pi} \int_{0}^{\pi} I \left(R^2 \sin{\theta} \ d\theta \ d\phi \right) = \int_{0}^{2 \pi} \int_{0}^{\pi} 25\,\sin\left(\theta\right)\,\left(25\,{\cos\left(\phi\right)}^2\,{\sin\left(\theta\right)}^2+{\left(5\,\sin\left(\phi\right)\,\sin\left(\theta\right)-1\right)}^2+{\left(5\,\cos\left(\theta\right)-2\right)}^2\right)\ d\theta \ d\phi$

Further simplifications left out as each term is expanded and the integral is split further as limits of integration are constant. The integral evaluates to $\boxed{3000 \pi}$ . The answer is, therefore, $\boxed{750}$ .