In my calculus class we learned about line integrals, and for homework we have exercise to find work done by gravitational force on material dot with mass $m$ which follows path of the elipse $\frac{x^2}{a^2}+\frac{z^2}{c^2}=1$ in second quadrant in positive direction (clockwise).
If I understood physics part of problem, gravitational force field is $f(X)=(0,0,-mg)$ and it's conservative which means work only depends on change in $z$ coordinate. And using formula with dot product I got: $W=\int_C-mgdz$ and since parametrization of curve $C$ is $x=a \cos t, z=c \sin t, t\in [\frac{\pi}{2},\pi]$ I have $W=-mgc$. It makes sense to me, but I'm not sure weather my conclusions are right. Can you check please and correct my mistakes if I have them.
Yes, your conclusion is correct. Since gravitational force is conservative in nature, work done by it is simply $-mgc.$