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

Well if you have 20 bats you probably want 20 balls? you don’t have enough context to answer the question.