Answer:
-937.5π
Step-by-step explanation:
F (r) = r = (x, y, z) the surface equation z = 3(x^2 + y^2) z_x = 6x, z_y = 6y the normal vector n = (- z_x, - z_y, 1) = (- 6x, - 6y, 1)
Thus, flux ∫∫s F · dS is given as;
∫∫ <x, y, z> · <-z_x, -z_y, 1> dA
=∫∫ <x, y, 3x² + 3y²> · <-6x, -6y, 1>dA , since z = 3x² + 3y²
Thus, flux is;
= ∫∫ -3(x² + y²) dA.
Since the region of integration is bounded by x² + y² = 25, let's convert to polar coordinates as follows:
∫(θ = 0 to 2π) ∫(r = 0 to 5) -3r² (r·dr·dθ)
= 2π ∫(r = 0 to 5) -3r³ dr
= -(6/4)πr^4 {for r = 0 to 5}
= -(6/4)5⁴π - (6/4)0⁴π
= -937.5π
