Final Answer:
a. The evaluation of the double integral ∬D (6x - 5y) dA over the region D, bounded by the circle with center at the origin and radius 1, results in 0.
Step-by-step explanation:
a. To evaluate the double integral ∬D (6x - 5y) dA over the region D, bounded by the circle with center at the origin and radius 1, we use polar coordinates. The region D in polar coordinates is described by and
The integrand (6x - 5y) is expressed in terms of polar coordinates as
The double integral evaluates to 0 due to the symmetry of the region and the cancellation of sine and cosine terms over the full circle.
Understanding the use of polar coordinates in double integration is essential for problems with circular symmetry. The substitution of variables transforms the rectangular region into a simpler form, allowing for easier integration. The symmetry of the region simplifies the integrand, leading to certain terms canceling out.
In summary, the double integral ∬D (6x - 5y) dA over the region D, bounded by the circle with center at the origin and radius 1, evaluates to 0. This result is achieved by exploiting the circular symmetry of the region in polar coordinates, demonstrating the efficiency of this coordinate system in certain integration problems.