Question

can you show me the work for 2,3,4,5, thank you

2. Evaluate ff curl F n dS, where F = (a2yz, yz2, 23e#v), and S is the part of the sphere a2 + y2+225 that lies above the plane z 1, oriented upwards. - Solution: -4T 3. A metal sheet is bent into the shape of the parabaloid r = y2+ 2 where 0 (r, y, z) is 6(x, y, z) = z. Find the mass of the resulting metal bowl. 1. The mass at Solution: (25 5+1) 4. Evaluate fe F. dr, where F = (axy, yz, za, and C is the triangle with vertices (1,0,0), (0, 1,0), and (0,0, 1), oriented clockwise when viewed from above. Solution: - 5. Calculate ffs F.n dS, where F (x3, y, 23 ) and S is the surface of the solid bounded by the cylinder r2y 1 and the planes z 0 and z 2.

Answer 1

by stokes theorem curlF Nds Fdr _ 'yzi+ yzj+z'ek -y2 = 4,z 1 where F and c is x+ Fdr x*yzdx+yzdy+ze" dz onc we have 2cost,y 2sin t,z = 1 dr2sint dt,dy 2cost dt,dz 0 and t> 0to 2T 2 a (4 cos(2 sint)(-2 sint)+(2sin)(2 costt -16sin2tcos* t-0L 4sint cost _ 2 a (2)+2 sin (2t)dt -4sin 1-cos4 r2a 2 sin (2 dt -4 -0 2 = 2 cos4-2+2sin (2tdt 2sin4t 2 cos 2 2t _ 4 2 =(0-47T-1)-(0-0-1) -47T _