The double integral
solve by reversing the order of integration
sketch the region
Did you sketch the region defined by the integral terminals. It's the region inside the triangle bounded by the x-axis and the lines x = 1 and y = 2x.
it should now be clear that your inetrgal can be written as
$\displaystyle \int_{x=0}^{x=1} \int_{y = 0}^{y = 2x} y \, e^{x^3} \, dy \, dx = \int_{x=0}^{x=1} e^{x^3} \, \int_{y = 0}^{y = 2x} y\, dy \, dx $.
Now integrate with respect to y. To do the integarl with respect to x, you should make an obvious substitution.