Here are the answers to each question on your Math 2415 Concept Quiz:

1. Formula for the integral $\int_C f \, ds$: The line integral of a scalar function $f$ over a curve $C$ is given by: $\int_C f \, ds = \int_a^b f(x(t), y(t), z(t)) \, \| \mathbf{r}'(t) \| \, dt$ where $\mathbf{r}(t)$ parametrizes the curve $C$ and $t$ varies from $a$ to $b$.

2. Formula for the integral $\int_C \mathbf{F} \cdot d\mathbf{r}$: The line integral of a vector field $\mathbf{F}$ over an oriented curve $C$ is given by: $\int_C \mathbf{F} \cdot d\mathbf{r} = \int_a^b \mathbf{F}(\mathbf{r}(t)) \cdot \mathbf{r}'(t) \, dt$ where $\mathbf{r}(t)$ is a parametrization of $C$ over the interval $[a, b]$.

3. Effect of reversing the orientation of $C$ in $\int_C \mathbf{F} \cdot ds$: If you reverse the orientation of the curve $C$, the line integral $\int_C \mathbf{F} \cdot ds$ changes sign. Therefore: $\int_{-C} \mathbf{F} \cdot ds = -\int_C \mathbf{F} \cdot ds$ where $-C$ denotes the curve $C$ with the opposite orientation.

Would you like more detailed explanations on any of these?

Here are five related questions that expand on this topic:

1. What is the difference between scalar and vector line integrals?
2. How do you find the parametrization of a curve for line integrals?
3. What is the physical interpretation of $\int_C f \, ds$ in applications?
4. How does Stokes' Theorem relate to line integrals of vector fields?
5. What conditions make $\int_C \mathbf{F} \cdot d\mathbf{r}$ path-independent?

Tip: For line integrals involving vector fields, remember that reversing orientation inverses the direction, which impacts the integral’s sign.