Question

The function f(x, y, z) = e^xyz is integrable over [0, 1] X [0, 1] X [0, 1]. is it true or false? justify

Answer 1

true.
we can just do integration...  as the given function can be broken into x,y,z parts separately.

f(x,y,z) can be expressed as F(x) * G(y) * H(z)    independent functions. So we can integrate the given function.

$\int \limits_{z=0}^1 \int \limits_{y=0}^1 \int \limits_{x=0}^1 {e^{xyz}} \, dx \, dy \, dz\\\\\int \limits_{z=0}^1 \int \limits_{y=0}^1 \int \limits_{x=0}^1 {e^{x}e^{y}e^{z}} \, dx \, dy \, dz\\\\\int \limits_{z=0}^1 {e^z} \, dz \ \int \limits_{y=0}^1 {e^y} \, dy \ \int \limits_{x=0}^1 {e^{x}} \, dx \\\\e^z)^1_0*(e^y)_0^1*(e^x)_0^1\\\\=(e-1)^3$