1. The area on sea-level of where the gold is located, this being the region R on our xy plane.
2. The signed height of each point in this region as given by . Positive corresponding to above sea level and negative to below sea level.

Our task is then very simple. We are to calculate the volume of space which occupies the location of our gold and decide whether to use ropes if we are venturing to the mountains or cranes if it is to the caves.

 or

Throw away all this fancy language we have ourselves a double integral problem. Some thinking should bring you to the idea that the double integral of this height function, our usual function is z, is interpreted as the volume of space above or below sea-level. Bear in mind that the result gives a signed volume.

We are given that information that , so setting the problem out formally, we are to calculate

bounded between the curves, ,  and , as shown below. Let’s get right to it.

As I mentioned in the previous videos, it is not essentially important that we graph out . We know that the double integral of this function gives us the signed volume. So in a way, the evaluation of the integrals will already ‘take care’ of the sign, and hence the interpretation. What is more important is how we define region R.

We see from our xy-plane graph that while the outer most limits of region R are -2 and 1, there are still points of intersection we need to handle. We cannot simply use one double integral and integrate between -2 to 1 because in the boundary , we would have included areas which are NOT inside the region R. The points of intersection  and  suggest to us we need to use different double integrals to evaluate . A solution is to break up region R into separate sub-regions.

,

and now we will consider these regions separately. Here goes.

Be careful when setting the x and y limits, more so for y limits for each double integral. They are different for each sub-region. All one needs to do is to apply the 2-step process, sweeping a point on a vertical line up and down then shifting the line left and right, to find the limits.

We end up with a rather simple but tedious integral to handle. Nonetheless, some careful calculations and you’ll get

Not such a nice number, after all who said the volume of caves would be ‘nice’ whole numbers. Still, just leave it as it is. Doing the same for the other sub-regions, we get

In case you missed it, I want to point out two quick facts about the x and y limits. The x limits must continue nicely from each region. Notice here that the upper x-limit of  joins up with the lower x-limit of . Also upper y-limit for all three sub-regions are the same. This is no surprise as our whole region R is bounded at the top by the same curve .

So after all our calculations, we see that summing the  terms will cancel each other, leaving us with

There we have it. We now know that the volume of the location of where the gold is held is negative meaning that it is probably in a cave and so we should bring along cranes to dig out the ground and find our gold.