Verified answer

Hola

Volumen entre los planos

x = 3 ; y = 3

x = 0 ; y = 0 ; z = 0

con la esfera

x^2 + y^2 + z^2 =64

como parte superior

Integral

V = ʃʃ [x_de_0_a_3] [y_de_0_a_3] (64 - x^2 - y^2)^(1/2) dx dy

Recordamos que

ʃ(a^2 - u^2)^(1/2) du = (1/2) u (a^2 - u^2)^(1/2) +

+ (1/2) a^2 asen(u/a) + C

Integramos con respecto a 'y'

tomando

u = y

a^2 = 64 - x^2

V = ʃ [x_de_0_a_3]

{ (1/2) y (64 - x^2 - y^2)^(1/2) +

+ (1/2) (64 - x^2) asen(y/(64 - x^2)^(1/2)) } dx [y_de_0_a_3]

V = ʃ [x_de_0_a_3]

{ (1/2) 3 (64 - x^2 - 3^2)^(1/2) +

+ (1/2) (64 - x^2) asen(3/(64 - x^2)^(1/2)) }

- { (1/2) (0) (64 - x^2 - 0^2)^(1/2) +

+ (1/2) (64 - x^2) asen(0/(64 - x^2)^(1/2)) }

dx

V = (3/2) ʃ [x_de_0_a_3] (55 - x^2)^(1/2) dx +

+ (1/2) ʃ [x_de_0_a_3] (64 - x^2) asen(3/(64 - x^2)^(1/2)) dx

La segunda integral no es elemental...

Ya te respondieron.

Y la calculadora??