Step 3 Using F(x, y, z) = 3y cos(z)i + e* sin(z)j + xelk, we have the following. 2 75 2T F(r(t) ) . r'(t ) at 2x -48 sin (t) -48 sin? (t) + 0 + 0) at Step 4 Remembering that sin?(t) = = - - cos(2t), we conclude that 48 / sin ?(1) at = =48 [- -cos(1) x -48nt Submit Skip (you cannot come back) Need Help? Read It Submit Answer 10. [4/6 Points] DETAILS PREVIOUS ANSWERS SCALC9 16.8.AE.001. MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER EXAMPLE 1 Evaluate S F . dr, where F(x, y, z) = -2y i + 3xj + 5z k and C is the curve of the intersection of the plane y + z = 2 and the cylinder x" + y" = 1. (Orient C to be counterclockwise when viewed from above.) SOLUTION The curve C (an ellipse) is shown in the figure. Although JF . dr could be evaluated directly, it's easier to use Stokes' Theorem. We first compute D OL curl F = -2v ( 3 + 4y Video Example () 3 x 572 Although there are many surfaces with boundary C, the most convenient choice is the elliptical region S in the plane y + z = 2 that is bounded by C. If we orient S upwards, then C has the induced positive orientation. The projection D of S on the xy-plane is the disk x2 + y's 1 and so with z = g(x, y) = 2 - y, we have SF . ar = / / curlF . as - / / 3 + 4y dA Jo . 13 + Arsin (0) Ir dr de lo de X )de - (27 ) + 0 = 37 Need Help? Read It Submit