True-False Determine whether the statement is true or false....

True-False Determine whether the statement is true or false. Explain your answer. (In Exercises $16-18$, assume that $C$ is a simple, smooth, closed curve, oriented counterclockwise.) It must be the case that
$$
\int_{C} e^{x^{2}} d x+\sin y^{3} d y=0
$$

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Key Concepts

Fundamental Theorem for Line Integrals

The fundamental theorem for line integrals states that if a vector field is conservative, then the line integral over a curve depends solely on the values of a potential function evaluated at the endpoints of the curve. Therefore, if the curve is closed, meaning that the endpoints coincide, the line integral must equal zero.

Simply Connected Regions

A simply connected region is one in which any closed curve can be continuously shrunk to a point without leaving the region. This condition is critical when applying tests for conservativeness and the fundamental theorem for line integrals. If the domain of a vector field is simply connected and the curl of the field vanishes, then the field is conservative.

Conservative Vector Fields

A vector field is called conservative if it can be expressed as the gradient of a scalar potential function. This property implies that the line integral of the field between two points depends only on the endpoints and not on the path taken. In a conservative field, the work done along any closed curve is zero, provided that the domain is simply connected.

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