I found $\displaystyle {Area}\quad =\quad \frac { 1 }{ 2 } \int { \overrightarrow { r } } \times \quad d\overrightarrow { r } $ as an equation applied in physics to find the area enclosed by a non-planar loop. I've been trying to verify the integral, and I'm having tons of trouble even beginning to evaluate it. Attached is a picture of all the work I've attempted thus far.

Thanks

Anthony