The question involves evaluating a line integral over a vector field along a path defined by a combination of equations. The integral is to be expressed in terms of a single variable through parametrization or direct substitution and solved as a definite integral.
The student is asking how to evaluate a line integral over a vector field along a specified path. The vector field is given by ∮C(3x²−8y²)dx+(4y−6xy)dy, where C is the curve defining the boundary of a region. Typically, when evaluating such line integrals, we parametrize the path using a suitable parameter, such as t, or directly substitute the equations defining the curve into the line integral, converting it into a definite integral in terms of a single variable, either x or y.
This process often involves finding expressions for both dx and dy in terms of the single variable chosen and then integrating the result over the appropriate limits.