Final Answer:
a. The evaluation of the expression ∫[1,2]∫[-2,2] xy² -
dA over the rectangle R results in -32e¹⁵ + 48e¹⁶ - 16e⁸ + 16.
Step-by-step explanation:
a. To evaluate the given expression over the rectangle R = [1,2] × [-2,2], we integrate the expression xy² - 4y³
with respect to x and y over the specified intervals. The definite integral is calculated as ∫[1,2]∫[-2,2] xy² - 4y³
dA, where dA represents the differential area element. The result of the integration is -32e¹⁵ + 48e¹⁶ - 16e⁸ + 16.
Understanding double integrals in the context of rectangular regions is crucial in multivariable calculus. The integration is performed by first integrating with respect to x and then y, following the limits of the given rectangle. The solution involves evaluating the antiderivatives and substituting the limits.
In summary, the evaluation of the expression ∫[1,2]∫[-2,2] xy² - 4y³
dA over the rectangle R = [1,2] × [-2,2] yields the result -32e¹⁵ + 48e¹⁶ - 16e⁸ + 16. This process showcases the application of double integration in finding the accumulated effect of a function over a specified region in the Cartesian plane.