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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 $.

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From $y \le x$ and $x \le 2$ you have $y \le 2$, so $0 \le y \le 2$ and $y \le x \le 2$.

$$\int_{0}^{2} \int_{y}^2 xy^2 \;dx \;dy $$

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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: $$I=\int_{0}^{2} \left(\int_{0}^x xy^2 \;dy \right)\;dx=\frac{32}{15}$$

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