I 15 points Find the volume of the solid that lies between t

I. (15 points) Find the volume of the solid that lies between the parabloid z = x2 + y2 and the sphere 2+y2+2 2 2. (20 points) Find the work done as a particle moves counter clockwise along a curve which forms a triangle between the points (1,0,0), (0,1,0), (0,0,1), under the influence of the force field F(x, y, z) = (z +92) i + (y + z*) j + (z + x*) k (a) Compute the work directly using line integrals of the for Pd+Qdy+Rdz or $F.dr (b) Compute the work using Stokes\' Theorem 3. (15 points) Use Stoke\'s Theorem to evaluate F.dr. Cis oriented counterclockwise as viewed from above. F (z, y, z) = i + (z + yz)j + (zy-v ), k. C is the boundary of the part of the plane 3x + 2y + z = 1 in the first octant 4. (20 points) Consider the following line integral srydx +r\'y* dy where C is the triangle with the verticies (0,0), (1,0), and (1,2), oriented clockwise (a) Evaluate the line integral using the form PQdy or F dr (b) Is the vector field F conservative? Why is it helpful to check for a conservative field? (c) Now evaluate using Green\'s Theorem 5. (15 points) Find a function such that F- and evaluate F·dr along the given curve C using the fundamental theorem of line integrals F (z, y, z) = (1 + xy)e(ry) + xe(r) j r(t) = Cos(t) i + 2 Sin(t) j, 0

Solution

F[X,Y] = [E^X]*[SIN(Y)] = P[X,Y] =C0+C1X+C2Y + C3XY+C4X^2+C5Y^2 P[Xi , Yi ] = F[Xi,Yi] …...AT ………..0 < = I <= 5 THE LINEAR SYSTEM IS TABULATED BELOW IN MATRIX FORM A = COEFICIENT MATRIX = UNKNOWNS F[X,Y] = POINT-I X(I) Y(I) C0 C1 C2 C3 C4 C5      C = [E^X]*[SIN(Y)]   0 0 0 1 0 0 0 0 0 C0 0 1 0 2 1 0 2 0 0 4 C1 0.9093 2 1 0 1 1 0 0 1 0     * C2       = 0 3 1 2 1 1 2 2 1 4 C3 2.4717 4 2 1 1 2 1 2 4 1 C4 6.2177 5 2 3 1 2 3 6 4 9 C5 1.0427 A*C = F ……………..C = [A INVERSE] * F A INVERSE = 1 0 0 0 0 0 -1.125 -0.375 1.25 0.75 -0.375 -0.125 0 0 -1 1 0.5 -0.5 0.5 -0.5 -0.5 0.5 0 0 0.125 0.375 -0.25 -0.75 0.375 0.125 -0.25 0.25 0.5 -0.5 -0.25 0.25      C = [ A INVERSE * F ] = C0       = 0 C1       = -0.94916 C2       = 5.05919 C3       = 0.78121 C4       = 0.94916 C5       = -2.30227 ANSWER………………