Evaluating the improper integral $\int_0^\infty \int_0^\infty e^{-(2x+7y)}\mathop{\mathrm dy} \mathop{\mathrm dx} $

Question

Evaluate the given improper integral

$$\int_0^\infty \int_0^\infty e^{-(2x+7y)}\mathop{\mathrm dy} \mathop{\mathrm dx} $$

Here is what I tried:

$$\int_0^\infty\lim\limits_{t \to \infty}[e^{-2x-7y}]_0^t \mathop{\mathrm dx}=\lim\limits_{t \to \infty} \int_0^t \frac{1}{7}e^{-2x}\mathop{\mathrm dx}= -\frac{1}{7}$$

Answer 1

Hint: this problem actually reduces to calculating

$$\left(\int_0^{\infty}e^{-2x}\,dx\right)\cdot\left(\int_0^{\infty}e^{-7y}\,dy\right)$$

Why?