MATH 215 Final: finalw14sol
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W2014, math 215 calculus iii, final exam solutions: (10 points) let. ~f (x, y, z) = (cos z + xy2)~i + xe z~j + (sin y + x2z)~k. Let e be the solid region bounded by the surface z = x2 + y2 and the surface z = 8 x2 y2. S be the boundary surface of e. evaluate the ux of ~f across s. (sol) use divergence theorem. Since div ~f = y2 + x2, the ux equals rrs. Note that the surfaces intersect when x2 + y2 = 4, z = 4. Thus the projection of e on the xy-plane is the disk x2 + y2 4. Using the cylindrical coordinates, the triple integral becomes. ~f d ~s = rrre r2 r dzdrd = 64. 0 r 8 r2 r2 (y2 + x2)dv . 3 : (10 points) let c be the curve from (0, 0) to (3, 27) along y = x3.
