Final answer:
The question involves sketching the region of integration for a given plane region R and setting up the integrals for both orders of integration, then evaluating the integral of ex² over R. The bounds form a trapezoidal shape, and the most convenient order might be to integrate with respect to x first due to the nature of the function ex².
Step-by-step explanation:
The student has been tasked with sketching the region of integration and setting up integrals for both orders of integration for the given plane region R, which is bounded by 0≤y≤18, x=y/3, and x=6. To sketch this region, we can note that the bounds form a trapezoidal shape with one vertical side (x=6), one slanted side following the line x=y/3, and the horizontal bounds being y=0 and y=18. The first order of integration would be with respect to y first (from x*3 to 18) and then with respect to x (from y/3 to 6). The second order of integration would be in x first (from y/3 to 6) and then in y (from 0 to 18).
Considering the integral ∫R ex² dA, the most convenient order of integration is often that which allows for simpler integration limits and potentially easier integration functions. Given the exponentiation of an x term, one might often choose to integrate with respect to x first. However, due to the nature of the exponential function and the fact that ex² does not have a simple antiderivative in terms of elementary functions, evaluating this integral directly may not be straightforward. In practice, this integral might require special functions or numerical methods for exact evaluation.