The problem you're working on seems to involve evaluating an integral with a change of variables. The integral is over a trapezoidal region $R$ with the given vertices and involves a function $7 \cos\left( 9 \left( \frac{y}{x+y} \right) \right)$.

To start solving this, we need to analyze the geometry of the trapezoidal region $R$ defined by the vertices $(3, 0), (8, 0), (0, 8), (0, 3)$. The task is to perform a change of variables, likely simplifying the region into a more standard form such as a rectangle, for easier integration.

Do you want help with how to set up the change of variables or further details about how to proceed with this type of problem?