Preview Activity 13.11.1.
In Preview Activity 13.5.1 we looked at how to understand line integrals of scalar functions through the analogy of running a mining machine along a given path. In short, the amount of copium mined depended on the density of copium at points on the path and the length of the path driven by the mining rig.
The opening tasks of Preview Activity 13.5.1 had you estimating the amount of copium mined by driving along the edge of a plot of land (drawn in yellow on Figure 13.11.1). This iterpretation meant that the scalar function we were using measured the linear density of copium. Thus, the Riemann sum we computed was the product of linear density of copium with the distance traveled.
(a)
In this task, we will interpret Figure 13.11.1 as a contour graph of the density of copium per unit area. This will allow us to compute the total amount of copium in our mining area by setting up a double integral. To approximate the total amount of copium available in our mine, do the following:
(i)
Break the mining plot (the area inside the yellow segments of Figure 13.11.1) into three pieces. Estimate the area of the three pieces you are using. Write a few sentences explaining your methods of estimating the areas.
(ii)
For each of the three pieces you used in part a.i, estimate the average density of copium on the piece. Write a few sentences explaining your methods of estimating the average density on each piece.
(iii)
Give an estimate for the total amount of copium on the mining plot and explain your computation.
(b)
What if instead of your mining plot being on a flat piece of land as represented in Figure 13.11.1, your mining plot was on a hill as represented in Figure 13.11.2. If we had the same copium density plot as a function of the \((x,y)\) coordinates, which of the following would you expect to be true?
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the total amount of copium available on Figure 13.11.2 is greater than on Figure 13.11.1
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the total amount of copium available on Figure 13.11.2 is lesser than on Figure 13.11.1
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the total amount of copium available on Figure 13.11.2 is the same as on Figure 13.11.1
Explain your reasoning for your choice.
