Final answer:
To evaluate the double integral with the given bounds, you can reverse the order of integration and integrate step by step, first with respect to y and then with respect to x.
Step-by-step explanation:
To evaluate the double integral ∫0∫π∫y∫π xsin x dx dy, it's possible to reverse the order of integration and then perform the integration step by step. To do this, we first need to consider the bounds for y and x after reversing the order of integration. Since y varies from 0 to π and x varies from y to π in the original order, after reversing, x would vary from 0 to π and y would vary from 0 to x. The new integral would be ∫0∫π∫0∫x xsin x dy dx. The integration proceeds by computing the inner integral with respect to y, which essentially turns into a multiplication by y, since xsin x is not a function of y. Completing the integration with respect to y yields x^2sin x / 2. Lastly, we would integrate x^2sin x / 2 with respect to x from 0 to π to find the final result.