In general, I know that for the triple integral $$\iiint f(x,y,z)dxdydz$$ the order of integration cannot be changed arbitrarily. However, if we know that the bounds are constants and that $$f(x,y,z)=f(x)g(y)h(z)$$ then can we rewrite the triple integral as $$\int f(x)dx \cdot \int g(y)dy \cdot \int h(z)dz$$ with the respective bounds?
I've seen several questions on MathStackExchange regarding change of order for triple integrals with nonconstant bounds, but I just wanted to verify that this is correct because I've encountered this type of integral several times in my physics course. Thanks!
Yes, $\int_a^b\int_c^d\int_s^t f(x)g(y)h(z) dzdydx= \int_a^b\int_c^d\left(\int_s^t f(x)g(y)h(z)dz\right)dydx$ and, since f(x) and g(y) do not depend upon z, that is the same as $\int_a^b\int_c^df(x)g(y)\left(\int_s^t f(x)g(y)h(z)dz\right)dydx$. Now, $\int_s^t h(z)dz$ is a number, call it "P", so we can factor it out of the integral and write $P\int_a^b\int_c^d f(x)g(y) dydz= P\int_a^b f(x)\left(\int_c^d g(y)dy\right)dx$ and do the same thing again.