- 105 Mixed Integral Theorems In A B D And E Take Normal Pointing Outwards From The Surface By Applying An A 1 (122.14 KiB) Viewed 16 times
105. Mixed Integral Theorems. In (a), (b), (d), and (e) take normal pointing outwards from the surface. By applying an appropriate integral theorem, evaluate the following integrals. Jl. (V × F)·dS over the portion of the surface 2z = x² + y² below the plane z = 2 when F(x, y, z) = (3y, -xz, —yz²). (b) ffi FdS where S is the surface of the closed cylinder x² + y² ≤ 1, 0≤z≤ 1 and F(x, y, z) = (1, 1, z(x² + y²)²). (c) [2³dy - y'da where C is the unit circle a² + y² = 1 traversed in the clockwise direction. (d) [[¸ F F. ds where F(x, y, z) = (2xy + z, y², −x − 3y) and S is surface of the tetrahedron - S bounded by 2x + 2y + z = 6, x = 0, y = 0, z = 0. (e) ff.s (▼ × F) · dS where F(x, y, z) = (zx + z²y + x, z³yx + y, z¹x²). Let S be the capped cylindrical surface given by the union of two surfaces S₁ and S₂ where S₁ is x² + y² = 1,0 ≤ z ≤ 1 and S₂ is x² + y² + (z − 1)² = 1, z ≥ 1.