Math 6B
Worksheet 7
Winter 2025
Due Monday, Feb 24, at 11:59pm.
1. Evaluate the line integral
where
(a) C is the circle x2 + y2 = 1.
(b) C is the circle (x — 1)2 + (y — 1)2 = 1.
2. Evaluate the work done by the force field F(x, y) = ⟨x, x2 + 3y2⟩ on an object moving along the straight line segments (0, 0) → (4, 0) → (2, 4) → (0, 0), which is a triangle.
3. Let R(x1, x2, x3) = 2x1i + x1x2 2 j + x1x2x3 k and S be the surface boundary of the solid bounded by x12 + x22 = 1, x12 + x22 = 4, x3 = 0, and x3 = 4. Evaluate the flux of R out of S.
4. Evaluate (x12 + x22)dS, where S is that section of the paraboloid x3 = 2(x12 + x22) between x3 = 0 and x3 = 4, along with the disc x12 + x22 ≤ 2, x3 = 4, oriented outwards.
5. Let R(x1, x2, x3) = x1 2 i + x2 2 j + x3 2 k, and C be the boundary of the circle x12 + x22 = 4 with x3 = 4 and CCW orientation when viewed from the origin. Evaluate the circulation R · dr.
6. Let F(x1, x2, x3) = (2x1 + x2) i + (2x2 — x1)j, and C be the helix C(t) = ⟨cost, sint, t⟩, t ∈ [0, 3π], along with the long segment from (—1, 0, 3π) to (1, 0, 0). Evaluate the circulation F · dr.