Use the region $\mathrm{R}$ with the indicated boundaries to evaluate the double integral. \[ \iint_{R} 4 x^{3} y d y d x ; \quad R \text { bounded by } y=x^{2}, y=2 x \] \[ \iint_{R} 4 x^{3} y d y d x= \]

Step 1 ：First, we need to find the intersection points of the curves y=x^2 and y=2x. These points will give us the limits of integration for x.

Step 2 ：The intersection points are (0,0) and (2,4).

Step 3 ：Next, we integrate the function \(4x^{3}y\) with respect to y first, and then with respect to x.

Step 4 ：The result of the double integral over the region R bounded by y=x^2 and y=2x is \(\frac{64}{3}\).