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Vector Calculus Independent Study Unit 7 Sample Test

1. [25 points] Find the surface area of the portion of the cylinder

y^{2} + z^{2} = 9

that is above the rectangle 0 ≤ x ≤ 2, −3 ≤ y ≤ 3.

2. [25 points] Evaluate the surface integral

∫_{S}∫ x^{2}z + y^{2}z dS,

where S is the portion of the sphere x^{2} + y^{2} + z^{2} = 9 above the plane z = 2.

3. [25 points] Find the flux of the vector field

F⃗ (x, y, z) = (x, y, 2z)

through the surface described by z = 1 − x^{2} − y^{2}, z ≥ 0.

4. [25 points] A certain surface is described parametrically by

S⃗(u, v) = (u cos v, u sin v, u^{2}).

Convert this to an implicit description (one just mentioning x, y, and z), and also find a normal to the surface at u = \sqrt{2}, v = π/4.