233, Math Review for Final Test The nal test will be cummulative, and the previous tests have covered almost all of the material from the class. The only new material involves, surface integrals, parametrized surfaces, Stokes Theorem, and the Divergence Theorem. For review on previously tested material, you should see your previous tests and the review problems given for them. For review on new material, look again at the questions assigned for homework. The following questions are a sampling of problems from the new material that have appeared on nal tests in the past. Some copies of nal exams from the past are available in the usual location on your instructors website. 1. (a) Set-up the surface integral for the mass of the surface z = x2 + y 2 where z 4, given that the density function is d(x, y, z) = x2 z. (b) Find the ux of the eld F (x, y, z) = zi k through the surface z = x2 + y 2 with z 4 and oriented with upward unit normal. 2. Use Stokes theorem to evaluate
C
F dr where C is the triangle that is the boundary of
the plane 2x + y + 3z = 6 in the rst octant, and F (x, y, z) = 2yi + 3zj + xk. (The orientation of C is counterclockwise with respect to the upward unit normal). 3. (a) Use the theorem divergence to nd the ux of F (x, y, z) = (x + y)i + yj + (z 2 2z + 1)k through the cylindrical can-like surface: x2 + y 2 = 4, z = 0 and z = 5 oriented with outward unit normal. (b) Find the Flux through the surface as in (a) except with the top z = 5 removed. 4. Let S be the surface lamina z = x2 + y 2 where 0 z 4. (a) (5 pts) Set up the integral to nd the mass of the paraboloid-shaped surface lamina given by z = x2 +y 2 where 0 z 4, and the density at each point on the surface is (x, y, z) = x2 +z 2 .. (b) (5 pts) Set up the integral to nd the ux of the vector eld F(x, y, z) = xi + yj + zk through the surface S oriented with upward unit normal. 5. Let S be the sphere of radius 2 centered at the origin ori...