Question

Use the divergence theorem to compute the value of flux integral S F · dS, where F(x, y, z) = (y3 + 3x)i + (xz + y)j + [z + x4 cos(x2y)]k and S is the area of the region bounded by x2 + y2 = 1, x ≥ 0, y ≥ 0, and 0 ≤ z ≤ 1.

Answer 1

The value of the flux integral is: ∫∫_S F · dS ≈ 4.325 + 0.917 = 5.242 .

Solution using the Divergence Theorem:

Step 1: Calculate the divergence of F

The divergence of F is:

div(F) = ∂Fx/∂x + ∂Fy/∂y + ∂Fz/∂z = 3 + xz + x^32xysin(x^2y) + 1

Step 2: Define the region of integration

The region of integration S is the unit disc in the xy-plane bounded by x^2 + y^2 = 1, x ≥ 0, and y ≥ 0, with a height of 1 in the z-direction (0 ≤ z ≤ 1). We can rewrite this region using cylindrical coordinates:

0 ≤ θ ≤ π/2

0 ≤ ρ ≤ 1

0 ≤ z ≤ 1

Step 3: Apply the Divergence Theorem

The Divergence Theorem states that:

∫∫_S F · dS = ∫∫∫_V div(F) dV

where V is the enclosed volume and S is its boundary surface.

In cylindrical coordinates, the divergence theorem becomes:

∫_0^(π/2) ∫_0^1 ∫_0^1 (3 + r^2 cos(r^2θ) + r^5 sin(r^2θ)) * r * dz * dr * dθ

Step 4: Evaluate the integral

The integral can be separated and evaluated:

∫_0^(π/2) dθ ∫_0^1 (3z + z^2 cos(z) + z^6 sin(z)) dz ∫_0^1 r dr

= π/2 * (9/2 + 1/3 sin(1) + 1/42 sin(1)) * 1/2

= π/2 * (27 + sin(1) + 1/21)

Step 5: Simplify the final answer

Therefore, the value of the flux integral is:

∫∫_S F · dS ≈ 4.325 + 0.917 = 5.242