(Solution) - For each of the following evaluate C F T -(2025 Original AI-Free Solution)

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For each of the following, evaluate ?C F ? T ds.
a) C is the topological boundary of the two-dimensional region in the first quadrant bounded by x = 0, v = 0, and y = y/4 - x2, oriented in the counterclockwise direction, and F(x, y) = (sin(?x3 - x2), xy).
b) C is the perimeter of the rectangle with vertices (0, 0), (2, 0), (0, 3), (2, 3), oriented in the counterclockwise direction, and F(x, y) = (ey log(x + 1)).
c) C = C1UC2, where C1 = ?B1(0, 0) oriented in the counterclockwise direction, C2 = ?B2(0, 0) oriented in the clockwise direction, and F(x, y) = (f(x2 + y2), xy2), where f is a C1 function on [1, 2].