~ 1. A surface has the vector function R(u, v) = hu2 , uv, 1i, where 1 ≤ u ≤ 2 and 0 ≤ v ≤ 1. Evaluate the surface integral ZZ yz dS. x
(Emir)
S
p 2. Using the positive orientation of the surface S defined by z = 3 − x2 + y 2 , find the flux of F~ (x, y, z) = h−y, x, zi over the portion of S that lies above the (Salma)
xy-plane. Z √ 1 ~ ds if C = R(t) := h t, t, 1i, t ∈ [1, 4]. 3. Evaluate x4
(Dion)
C
4. Verify Green’s Theorem for the line integral I (y sec2 x − xy 2 ) dx + 4xy 2 dy C
where C is the boundary of the triangle with vertices (0, 0), (1, 1), and (0, 1). (Angelo[GT] and Ryan[Def’n]) 5. Consider the vector field F~ (x, y, z) = h2xy, x2 + z, y + 1i.
(Sherlyne)
(a) Determine divF~ . (b) Show that F~ is conservative. Z (c) Using the Fundamental Theorem of Line Integrals, compute
F~ · d~r
C
4t sin2 t 4t cos2 t ~ , , sec2 ti, t ∈ [0, π/4] where C is R(t) =h π π 6. Set-up an iterated triple integral that gives the volume of the solid in the first octant bounded by the cylinder x2 + y 2 = 9 and the plane x + y = 2. (Geoffrey) 7. Set-up an iterated triple integral in cylindrical coordinates that gives the mass of the solid bounded by the plane z = 2 and the paraboloid z = 4 − x2 − y 2 given that the density at every point is δ(x, y, z) = xyz. (John Mark) ZZZ z 8. Evaluate dV where R is the solid inside the sphere x2 + y 2 + 2 x + y2 + z2 R p 2 z = 4 and below the cone z = x2 + y 2 . (Patrick)