Compute $\int_C ze^{\sqrt{x^2+y^2}} \mathrm ds$ 
Compute $\int_C ze^{\sqrt{x^2+y^2}} \mathrm ds$ where
$$C:x^2+y^2+z^2=a^2, x+ y=0, a \gt 0$$

At first I thought to parametrize this as: $x=a \cos t , y=a \sin t, z =0$, but then the integral will result in $0$ and this might not be true.

$x=a \cos t , y=a \sin t, z =0$ is not a parametrisation of $C$ as $x + y \neq 0$. Try plugging $x+y=0$ into $x^2+y^2+z^2 = a^2$ and then use your approach with $\sin$ and $\cos$.
