5.6 Line Integrals of Scalar Functions (continued)

Properties of Scalar Line Integrals

Visualizations of Scalar Line Integrals as Area Under a Curve

Explain in your own words what $\displaystyle\int_C f \, ds$ means in the copium analogy and what exactly would be measured by this scalar line integral.

Explain in your own words what $\displaystyle \int_C (k f) \, ds = k \int_C f \, ds$ means in the copium analogy. It may be helpful to describe each side of the equation separately and say why they are equal in the analogy.

Explain in your own words what $\displaystyle\int_C (f+g) \, ds = \int_C f \, ds + \int_C g \, ds$ means in the copium analogy. It may be helpful to describe each side of the equation separately and say why they are equal in the analogy. You may also find it helpful to consider $f$ and $g$ to measure density of different materials.

Explain in your own words what $\displaystyle\int_{-C} f \, ds = \int_C f \, ds$ means in the copium analogy. It may be helpful to describe each side of the equation separately and say why they are equal in the analogy.

Explain in your own words what $\displaystyle\int_{C_1+C_2} f \, ds = \int_{C_1} f \, ds + \int_{C_2} f \, ds$ means in the copium analogy. It may be helpful to describe each side of the equation separately and say why they are equal in the analogy.

Let $C$ be the path given below from $P$ to $Q$ with pieces $C_1$, $C_2$, and $C_3$ as labeled. Let $f$ be a scalar-valued function such that $\int_C f \, ds = 13$, $\int_{C_1} f \, ds = 5$,and $\int_{C_3} f \, ds = 9$.

Find the following: -$\int_{-C_3} f \, ds$-$\int_{C_2} f \, ds$-$\int_{-C_1-C_3} f \, ds$

Calculate the following line integral where $C$ is the path on $x=y$ with $-1 \geq y \geq 2$: