Can you demonstrate this propiety of integrals?

Question

If f: R ^ 2 → R, defined by f (x, y) = g(x)h(y), where g and h are real functions of continuous real variable. If R = {(x, y) ∈ R^2: a ≤ x ≤ b, c ≤ y ≤ d}, prove that:
∫ ∫ f (x,y) d (x, y) = ∫ g(t) dt ∫ h(t) dt

(The intregral of ∫ g(t)dt is from a to b, the integral of ∫ h(t)dt is from c to d)

Thanks you ! :)

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Answer 1 · 2018-05-21T20:06:06

Proof below

First you need to believe in the area element.

#J = int int_R f * dA = int_c^d int_a^b g(x) * h(y) * dx * dy#

Then you need to see that x does not depend on y.

#J = int_c^d [int_a^b g(x) * h(y) * dx] dy#

#int_c^d K * dy = K int_c^d dy#

The constant goes to the left.

#J = [int_a^b g(x) * dx] * int_c^d h(y) * dy#