Again, for simplicity, we say that a line integral can be interpreted as the area above a path to some surface or ceiling. However, it is important to note that the accumulation function is about accumulating other derived quantities, not strictly area.
However, just for our example, we are thinking of this as finding the area of one side of the dam surface!
So let's simulate how we accumulate in 3D dimensions! Use the applet and move the slider to see how the line integral is accumulated! Here, we have already partitioned the arc into infinitesimal arc lengths ds, and we are integrating over that path!

The Aftermath...

Here is the uniquely shaped dam! Take a look!
Wow...cool...