Q: Let $E$ be the region of $\mathbb{R^3}$ bounded by the plane $z = 0$, the cylinder $y = x^2$, the cylinder $x = y^2$ , and the graph of $z = xy$. Find the volume of $E$.
I imagine that i can consider that $z$ varies from $0$ to $xy$, while $y$ varies from $0$ to $x^2$, and $x$ varies from $0$ to $1$.
Then the volume of $E$ is equal to:
$$\int\limits_0^1\int\limits_0^{x^2}\int\limits_0^{xy}\mathrm dz\,\mathrm dy\,\mathrm dx = \dfrac{1}{12}$$
But i'm starting to solve this integral questions and i wonder if i answered it correctly. Can anybody check, and if i'm wrong, help me to understand what i missed? Thanks!!
This set-up is nearly correct. The bounds for $y$ are not right.
How are you using the curve $x = y^2?$
At a minimum, $y = x^2$ at a maximum $y = \sqrt x$
$$\int\limits_0^1\int\limits_{x^2}^\sqrt x\int\limits_0^{xy}\mathrm dz\,\mathrm dy\,\mathrm dx$$