Activity 13.5.4. Explaining Properties of Scalar Line Integrals.
In this activity, we will be explaining each of the Properties from Properties of Scalar Line Integrals in the context of our copium mining analogy from Preview Activity 13.5.1. Remember that the curve in our scalar line integral coresponds to the path the mining rig will take and the function in the scalar line integral measures the density of copium at that point on the surface.
(a)
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.
(b)
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.
(c)
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.
(d)
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.
(e)
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.
