# Double integral integration limits

For multivariable function $f(x,y) = xy^2$ how to find the interval of integration, the

question gives those inequalities $x \le 2, y \ge 0, y \le x$.

From $y \le x$ and $x \le 2$ you have $y \le 2$, so $0 \le y \le 2$ and $y \le x \le 2$.
$$\int_{0}^{2} \int_{y}^2 xy^2 \;dx \;dy$$
There are two ways to do the integration. The first one is like in the first answer: $$I=\int_{0}^{2} \left(\int_{y}^2 xy^2 \;dx \right)\;dy=\frac{32}{15}$$
The second one is similar to the condition you specify in the original question: $$I=\int_{0}^{2} \left(\int_{0}^x xy^2 \;dy \right)\;dx=\frac{32}{15}$$