Final answer:
The question involves evaluating a line integral over a parabolic path and finding the area enclosed by a parametric curve using integration.
Step-by-step explanation:
The student is asking about evaluating a line integral of a function over a specified path, which in this case is the arc of a parabola. The problem involves using parameterization to express one variable in terms of another along a given curve, which simplifies the integral to a function of a single variable, thus making the integration more straightforward. For instance, along the parabolic path, one might solve for x in terms of y, or vice versa, to obtain an expression that can be integrated using standard techniques. Additionally, the student is asked to find the area enclosed by a parametric curve, which can be approached by integrating the given parametric equations over the interval from 0 to 2π.
To solve for the line integral, we would need to replace x and dy using the relationships derived from the parabolic equation, for example, y = x² and dy = 2x dx, and then integrate over the specified limits. The area enclosed by a curve defined by parametric equations can be found by integrating x dy - y dx over the given range of the parameter, which