Use stokes' theorem to evaluate c f · dr where c is oriented counterclockwise as viewed from above. f(x, y, z) = xyi + 2zj + 6yk, c is the curve of intersection of the plane x + z = 6 and the cylinder x2 + y2 = 144.

Stokes' theorem says



where is the region with boundary . We can parameterize this region by



with and . This gives us a surface element of



Meanwhile, the curl of the given vector field is




So the surface integral reduces to



You can verify this by computing the equivalent line integral using the parameterization



with . You should (obviously) get the same result.