Find the work done by the vector field f~ = x 2y~i + 1 3 x 3~j + xy~k along the curve of intersection c of the paraboloid z = y 2 − x 2 and the cylinder x 2 + y 2 = 1, oriented counterclockwise as viewed from the above.

Let and . Then can be parameterized by

with , and its derivative is

Now,

Then the work done by along is

∠acb = 8°

step-by-step explanation:

∠xba lies on the circle and is half the measure of the intercepted arc ax

hence

measure of arc ax = 2 × 42° = 84°

∠acb is outside the circle and is

∠acb = (arcab - arcax ), hence

∠acb = 0.5 (100 - 84) = 0.5 × 16° = 8°