Use stokes' theorem to evaluate c f · dr where c is oriented counterclockwise as viewed from above. f(x, y, z) = 2yi + xzj + (x + y)k, c is the curve of intersection of the plane z = y + 6 and the cylinder x2 + y2 = 1.

By Stokes' theorem, the line integral of over is given by the surface integral of the curl of over , where is the region of intersection of the plane and the cylinder with having positive/upward orientation.

Parameterize by

with and .

Take the normal vector to to be

The curl of is

Then the line integral is equivalent to

answer: 59

step-by-step explanation: