Evaluate the integral below, where c is the curver(t) = ‹sin(t),cos(t), sin(2t)›, 0 ≤ t ≤2π. (hint: observe that c lies on the surfacez = 2xy.)
∫c (y+9sin(x))dx +(z2+4cos(y))dy +x3dz

We can compute the integral directly: we have

Then the integral is

You could also take advantage of Stokes' theorem, which says the line integral of a vector field along a closed curve is equal to the surface integral of the curl of over any surface that has as its boundary.

In this case, the underlying field is

which has curl

We can parameterize by

with and .

Note that when viewed from above, has negative orientation (a particle traveling on this path moves in a clockwise direction). Take the normal vector to to be pointing downward, given by

Then the integral is

Both integrals are kind of tedious to compute, but personally I prefer the latter method. Either way, you end up with a value of .

mississippi is the answer

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