9.8.8 Surface integrals

Problem: Calculate the surface integral of the \( 2^{\text {nd }} \) kind: \[ \iint_{S} x^{3} d y d z+y^{3} d z d x \] where \( S \) is the outside of the ellipsoid part \[ \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1, \quad z \geq 0 . \]