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Date

Vector Calculus Independent Study

Unit 6 Sample Test

1. [25 points] Re-write

∫_{0}^{1} ∫_{x}^{1} ∫_{y}^{x} f(x, y, z) dz dy dx

in all five of the other orders of integration.

2. [25 points] Suppose a region in R^{3} is bounded by a surface described by the equation ρ = f(θ, ϕ) in spherical coordinates. Show that the volume enclosed by the surface is

V = \frac{1}{3} ∫_{0}^{2π} ∫_{0}^{π} f^{3}(θ, ϕ) sin ϕ dϕ dθ

3. [50 points] The temperature inside a rectangular box is described by 0 ≤ x ≤ 1, 0 ≤ y ≤ 2, 0 ≤ z ≤ 3 is kd^{2}, where d is the distance to the origin and k is constant. What is the average temperature in the rectangular box?

4. [25 points] Find the volume of the tetrahedron bounded by the coordinate planes and the plane z = 4 − 3x − 2y.