Double Integrals over Rectangles

Over a rectangle with constant limits, a double integral of a separable integrand splits into a product of two ordinary integrals: $\int_a^b \int_c^d f(x)g(y)\,dy\,dx = \left(\int_a^b f(x)\,dx\right)\left(\int_c^d g(y)\,dy\right)$.

So $\int_0^1 \int_1^2 xy\,dy\,dx = \left(\int_0^1 x\,dx\right)\left(\int_1^2 y\,dy\right) = \frac{1}{2} \cdot \frac{3}{2} = \frac{3}{4}$.

Do the two single integrals mentally and multiply. Never expand the inner integral symbolically first — that is where the time goes.