Question about double integrals

Question

Can someone explain to me what I'm doing wrong on this problem. I've been stuck on this problem for a while. Please help.

Sketch the region $D$ that gives rise to repeated integral and change the order of integration.

Below is my work. Where am I going wrong?

Answer 1

You have to divide the reverse integral in two pieces, notably

$$\int_{\frac12}^1 \int_{x^3}^{x} f(x,y)dydx=\int_{\frac18}^{\frac12} \int_{\frac12}^{\sqrt[3]y} f(x,y)dxdy+\int_{\frac12}^1 \int_{y}^{\sqrt[3]y} f(x,y)dxdy$$

Answer 2

The inner limit is always on the right should be $\sqrt[3]{y}$

However, the left limit is not always $x=y$, it is bounded on the left by $\max(\frac12, {y}).$