J. Evaluate the surface integral F. dS, where F(x, y, z) = xi+yj+zk is taken over S the paraboloid z = 1 – x2 – y², z > 0.

7. Verify Stokes' theorem by direct computation of both the line integral and the surface integral for Fi(yz + 2) + j(yz + 4) − kzy, where S is the surface of the cube x = 0, y = 0, z=0, x=2, y = 2, z = 2 above the xy-plane. 1

Use Stokes' theorem to calculate (V × A) · ndS with A = 3yi – xzj+ yz²k, where S is the surface of the paraboloid 2z = x² + y? bounded by z= 2 and C as the boundary.

2. Express the area of the given surface as an iterated double integral in polar coordinates, and then find the surface area. (a) the portion of the cone z =√x² + y² that lies inside the cylinder x² + y² = 2x. (b) the portion of the paraboloid z=1-x² - y2 that is above the xy-plane. (c) the portion of the surface z = xy that is above the sector in the first quadrant bounded by the lines y = x/√3, y = 0, and the circle x² + y²=9. (d) the portion of the paraboloid 2z = x² + y² that is inside the cylinder x² + y² = 8. (e) the portion of the sphere x² + y² +2²=16 between the planes z = 1 and z = 2. (f) the portion of the sphere x². + y² + z² = 8 that is inside the cone z = √x² + y²₁