Activity 13.11.3. Explaining Properties of Scalar Surface Integrals.
In this activity, we will be explaining each of the properties from Properties of Scalar Line Integrals in the context of a new analogy. We have just purchased a plot of land that spans two mountains, Sugar Mountain and Spice Mountain. We will label the plot of land on Sugar Mountain \(S_1\) and the plot of land on Spice Mountain \(S_2\text{.}\) Unfortunately there are two types of toxic organisms on the surface of your new land, which may explain why you paid so little for the land. Let \(f\) be the density of the toxic fungus on your new plot of land and let \(g\) be the density of toxic bacteria on the new plot.
(a)
Explain in your own words what \(\iint_{S_1} f \, dS\) means in the above analogy and what exactly would be measured by this scalar line integral.
(b)
Explain in your own words what \(\displaystyle \iint_{S_1} (2 f) \, dS = 2 \iint_{S_1} f \, dS\) means in the new 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 \(\displaystyle \iint_{S_2} (f+g) \, dS = \iint_{S_2} f \, dS + \iint_{S_2} g \, dS\) means in the new 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 \(\displaystyle \iint_{-S_2} f \, dS = \iint_{S_2} f \, dS\) means in the new 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 \(\displaystyle \iint_{S_1+S_2} f \, dS = \iint_{S_1} f \, dS + \iint_{S_2} f \, dS\) means in the new analogy. It may be helpful to describe each side of the equation separately and say why they are equal in the analogy.
