$\int_{0}^{2} \int_{x}^{2} 2 y^{2} \sin (x y) d y d x$

Step 1 ：First, we compute the inner integral, treating x as a constant. The inner integral is \(\int_{x}^{2} 2 y^{2} \sin (x y) d y\).

Step 2 ：Next, we substitute the limits of integration for y, which are x and 2.

Step 3 ：Then, we compute the outer integral, which is \(\int_{0}^{2} \text{{inner integral}} d x\).

Step 4 ：We substitute the limits of integration for x, which are 0 and 2.

Step 5 ：Finally, we simplify the result to get the final answer.

Step 6 ：The final answer is \(\boxed{4 - \sin(4)}\).