I need help with this question:

F= (x+y)i+(y^(2)+z)j+(z-x)k
find the work done by F along the path C. C consists of the straight line segment from (1,6,3) to (3,0,0) followed by the ellipse (x^2)/9 + (y^2)/4) = 1
in the xy plane starting at (3,0,0) and taken once in an anti clockwise diection when viewed from above.

-i think firstly I should calculate the work done along the straight line 1st and then the work done in the ellipse and then add them together...

for the straight line segment from the equation
r= -i+6j+3k+t(2i-6j-3k) I get x=1+2t y=6-6t z=3-3t
and dz=2 dy=-6 and dz=-3

and when I sub these into F and integrate from 1 to 0 with respect to dt I get -75.5. is that right?

then, I'm not sure what to do with the ellipse,
should I let x=t y=sqrt(4 - ((4t^2)/9)) and go from there? It looks a bit messy to do, is there a simpler way?