Evaluate c (y + 6 sin(x)) dx + (z2 + 4 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.) c f · dr =

You first have to observe that r(t) = sin t i+ cos t j sin 2t k is negative oriented

Using Stoke's theorem:

Lower limit: C

Integral of equation with lower limit of c (y = sin^3 x) dx + (z^2 + cos^4y) dy + (x^3 + tan^5z) dz = - double integral of lower limit S delta F ds

where:

S (surface z) = 2xy bounded by D = Open Bracket x^2 + y^2 less than or equal to 1 close bracket

Please refer to the image attached for the solution, exact details and the answer