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Volume Integral

Indefinite integral

integral integral integral(x^2 + y^2 + z^2) dx dy dz = c_1 x y + c_2 x + c_3 + 1/3 x^3 y z + 1/3 x y z (y^2 + z^2)

Definite integral over a cube of edge length 2 L

integral_(-L)^L integral_(-L)^L integral_(-L)^L (x^2 + y^2 + z^2) dx dy dz = 8 L^5

Definite integral over a sphere of radius R

integral integral integral_(x^2 + y^2 + z^2<R^2)(x^2 + y^2 + z^2) dz dy dx = (4 π R^5)/5