Verified answer

z = √(a² - x² - y²), o emisfério superio.

dS = √(1 + (z_x)² + (z_y)²) dA

.....= √(1 + (-x/√(a² - x² - y²))² + (-y/√(a² - x² - y²))²) dA

.....= √(1 + (x² + y²)/(a² - x² - y²)) dA

.....= √(((a² - x² - y²) + (x² + y²))/(a² - x² - y²)) dA

.....= (a/√(a² - x² - y²)) dA.

∫∫s f(x,y,z) dS

= ∫∫ x² * (a/√(a² - x² - y²)) dA

= ∫(θ = 0 to 2π) ∫(r = 0 to a) (r cos θ)² * (a/√(a² - r²)) * (r dr dθ), onde x = r cos θ, y = r sin θ

= ∫(θ = 0 to 2π) cos² θ dθ * ∫(r = 0 to a) ar² (a² - r²)^(-1/2) * r dr

= ∫(θ = 0 to 2π) (1/2)(1 + cos(2θ)) dθ * ∫(u = a² to 0) a(u+a²) u^(-1/2) * (-1/2) du, onde u = a² - r²

= ∫(θ = 0 to 2π) (1/2)(1 + cos(2θ)) dθ * (a/2) ∫(u = 0 to a²) (u^(1/2) + a² u^(-1/2)) du

= [(1/2)(θ + sin(2θ)/2) {θ = 0 to 2π}] * [(a/2) ((2/3)u^(3/2) + 2a² u^(1/2)) {u = 0 to a²}]

= 4πa³/3.

Boa sorte!