Finding $\iint_W x^2y\,\mathrm{d}x\,\mathrm{d}y$ where $W$ is a rectangle

Question

I'm learning double integrals and I'm trying to calculate the following integral:

$$\iint_{W} x^2y \,\mathrm{d}x\,\mathrm{d}y\,,$$ where $W$ is a rectangle given by points: $A=(0,1), B=(2,1), C=(2,2), D=(0,2)$.

Could you please help me calculate the following integral? I know how to calculate double integrals in general, I just don't know how to get the boundaries.

Thanks

Answer 1

Considering that the endpoints of the rectangle are $(0, 1)$, $(2, 1),$ $(2, 2),$ and $(0, 2),$ the rectangle can be viewed as the set of points $\mathcal R = \{(x, y) \,|\, 0 \leq x \leq 2 \text{ and } 1 \leq y \leq 2\}.$ (For instance, you can click this link to view the four endpoints in Geogebra.) Consequently, the integral is given by $$\int_1^2 \int_0^2 x^2 y \, dx \, dy.$$ Can you finish up by computing this double integral?