(a) Let C be the circle x + y^2 = 9, oriented counterclockwise. Use Green's Theorem to evaluate the following integral:
∫(sin(x^2 - y)dx + xy) dy
(b) Let S be the boundary of the region x + y ≤ 9, 0 ≤ z ≤ 4, oriented with unit normals pointing outwards. Consider the vector field:
F = (x + sin(y))i + yzj + (3yz + sin(xy))k
Use the divergence theorem to evaluate the following integral:
∫∫F · dA
(c) Let C be the rectangle in R^3 with vertices (0,0,0), (1,0,0), (0,1,0), (1,1,0), (0,0,1), (1,0,1), (0,1,1), and (1,1,1), oriented in the given order. Use Stokes's Theorem to evaluate the following integral:
∫∫(cos(x^2)dx + xydy + xzdz)
