Question
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
Explanation / Answer
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