Final answer:
The evaluation of the double integral ∫₀²∫ₓ²⁴ x cos(y²) d y d x uses Fubini's Theorem to confirm the order of integration can be swapped due to the continuity of the functions over the defined rectangle. Then, one should integrate first with respect to y, treating x as constant, and afterward with respect to x.
Step-by-step explanation:
The question asks for the evaluation of a double integral using Fubini's Theorem. The integral to be evaluated is:
∫₀²∫ₓ²⁴ x cos(y²) d y d x
By Fubini's Theorem, we know that we can interchange the order of integration if the two functions being integrated are continuous over the rectangle defined by the limits of integration. In this case, since x and cos(y²) are continuous functions, we can apply Fubini's Theorem to swap the order of integration.
To proceed with the evaluation, first integrate with respect to y while treating x as a constant, and then integrate the resulting expression with respect to x. This computation requires knowledge of integration techniques, such as integration by parts or recognizing standard integral forms.