Written Homework for June 26, 2024, Vector Calculus II First Name Last Name OSU ID Grading Some written homework problems are graded for correctness. To earn full credit for each problem you need to clearly and neatly show your work or provide a written explanation when appropriate. Partial credit will be awarded for logical, organized progress toward a solution when work is required and shown. Correct answers with no supporting work will not receive full credit. Problems 1. Let 0 ≤ a ≤ b and let f (x) ≥ g(x) on [a, b]. Let R be the region on the xz-plane between z = f (x) and z = g(x) over [a, b], and let D be the solid region obtained by rotating R around the z-axis. (a) Give D as a set of triples in cylindrical coordinates: D = {(r, θ, z) | ? ≤ r ≤ ?, ? ≤ θ ≤ ?, ? ≤ z ≤ ?} (b) Let V be the volume of D. Prove the shell formula: V = y dV = 2π Z b a D 1 x(f (x) − g(x)) dx 2. The temperature in a (very strange) room is given by the function w = f (x, y, z) = x2 y 2 z 2 . (a) There is a helix in the room with parameterization ⟨x, y, z⟩ = ⟨cos t, sin t, t⟩. Use the . Reflect on whether your answer makes sense. chain rule to find dw dt (b) There is a cone in the room with parameterization ⟨x, y, z⟩ = ⟨r cos θ, r sin θ, r⟩. Use the chain rule to find ∂∂rw and ∂∂θw . Reflect on whether your answers make sense. 2 3. The electrical voltage in some 3D region is given by V (x, y, z) = x3 − 5xy xyz. Consider P (2, 8, 5) and ⃗v = ⟨1, 1, −1⟩. (a) Find the rate of change of the voltage at P in the direction ov ⃗v . (b) In what direction does the voltage increase/decrease most rapidly at P ? What is the magnitude of the rate of change? (c) Give two orthogonal directions in which the rate of change of the voltage at P is zero. 3 √ 4−y 2 √ 2 x y 2 − 1 dx dy. 4. Use polar coordinates to evaluate R √3 R 5. Use polar coordinates to evaluate R ∞ R ∞ −x2 −y2 e dydx. 0 0 1 0 Bonus: how can we use this to compute the one-dimensional integral 4 R ∞ −x2 e dx? 0