Find the surface integral ∬_(S)F*ndA by Divergence Theorem
S: is the cube bounded by x=±2, y=±2, z=±2 and F=[x^(2), y^(2), z^(2)]
S: surface of a unit sphere centered at the origin in the first octant (x, y, z ≥ 0) and F= x^(2)y, xz, 2xyz
S is the surface of intersection of x^(2) + z^(2) = 4, x^(2) + y^(2) = 4, F=[x^(2)y, -y^(2)x, z^(2)]
[Hint: This is the Steinmetz Solid. The volume can be computed by
∫_(-2)^2 ∫_(-√(4-x^(2)))^(√(4-x^(2))) ∫_(-√(4-x^(2)))^(√(4-x^(2))) dzdydx = 128/3.]
S is the surface of the cone x^(2) + y^(2) ≤ 4z^(2), 0 ≤ z ≤ 3, F=[xy, yz, xz]
Find the surface integral ∬F · ndA by Divergence Theorem
1. S: is the cube bounded by x=2, y=2, z=2 and F=[x^(2), y^(2), z^(2)]
2. S: surface of a unit sphere centered at the origin in the first octant (x, y, z ≥ 0) and F=[x^(2)y, xz, 2xyz]
3. S is the surface of intersection of x^(2) + z^(2) = 4, x^(2) + y^(2) = 4, F=[x^(2)y, -y^(2)x, z^(2)] [Hint: This is the Steinmetz Solid. The volume can be computed by
4. S is the surface of the cone x^(2) + y^(2) ≤ 4z^(2), 0 ≤ z ≤ 3, F=[xy, yz, xz]