Compute Line Integral of Vector Field along a Curve

Compute the line integral of the vector field F on a curve $C$, using the parameterization $\mathbf{r}(t)$ from $t = a$ to $t = b$. The line integral is given by $\int_{a}^{b} \mathbf{F}(\mathbf{r}(t)) \cdot \frac{d\mathbf{r}}{dt} \, dt$.

In this problem, we are tasked with computing the line integral of a vector field over a parameterized curve. Understanding line integrals is crucial in fields such as physics and engineering because they allow us to compute work done by a force field along a path, among other applications. This type of problem represents a combination of vector calculus concepts, particularly focusing on integrating along curves within vector fields.

To tackle this problem, one must first be familiar with the concept of a vector field, which assigns a vector to each point in space. Additionally, it's important to understand parameterization of curves, where we express a curve using a parameter, often denoted as t, that allows us to traverse along the curve by varying t within a given interval.

The core concept in solving this problem involves applying the definition of a line integral. You transform the vector field into a function of the parameter t via the given parameterization, and then compute the dot product of this vector function with the derivative of the parameterization vector. The integration process involves solving this dot product over the specified range of the parameter. Mastery of these concepts will enable you to simplify and compute line integrals effectively in various contexts.