Evaluate c (y + 7 sin(x)) dx + (z2 + 9 cos(y)) dy + x3 dz where c is the curve r(t) = sin(t), cos(t), sin(2t) , 0 ≤ t ≤ 2π. (hint: observe that c lies on the surface z = 2xy.)

Treat as the boundary of the region , where is the part of the surface bounded by . We write



with .

By Stoke's theorem, the line integral is equivalent to the surface integral over of the curl of . We have



so the line integral is equivalent to




where is a vector-valued function that parameterizes . In this case, we can take



with and . Then



and the integral becomes