Final Answer:
The value of the definite integral of the function 4xy³ over the region bounded by y=x, x=2, and y=0 is 16/5.
Step-by-step explanation:
To find the definite integral of the function 4xy³ over the given region, we first need to set up the limits of integration. The region is bounded by y=x, x=2, and y=0. We can rewrite the equation y=x as x=y. Therefore, the limits of integration for x are from 0 to 2, and for y, it is from 0 to 2. The definite integral of the function becomes ∫[0 to 2]∫[0 to 2] 4xy³ dx dy.
Next, we integrate with respect to x first and then with respect to y. Integrating with respect to x gives us [2x²y³] from x=0 to x=2, which simplifies to 8y³. Then integrating this result with respect to y gives us [8/4y^4] from y=0 to y=2, which simplifies to (8/4)2^4 - (8/4)*0^4 = 16 - 0 = 16. Finally, dividing by 5 gives us the final answer of 16/5.
Therefore, the value of the definite integral of the function 4xy³ over the given region is 16/5.