The volume calculation computes the volume under the surface (dem) with map extent restricted to the user drawn box/polygon. The bottom of the volume calculated, the surface you would walk on if the volume was excavated or removed, is a flat plane at the lowest elevation within the user drawn polygon.
In 2D, the left panel, the area under the curve in pink would have a and b set by the user (the drawn polygon). Currently we shift the curve so that the lowest part within the pink zone would touch the x-axis, then we calculate the area between curve and the x-axis.
Expanded back into 3D, right panel, the blue surface would be the extent defined by the user drawn polygon. We then take the lowest point on the surface in blue and shift the surface down the z-axis so that the lowest point touches the xy-plane. Then we calculate the volume between the surface and the plane.
Route Tool Explanation
The route calculation uses a least cost path analysis to determine the shortest, most cost-effective route between the two end points. Cost is calculated as defined by the layers selected for the current map display. Emphasis for the least cost calculation is given to higher values (closest to 100% in the default Weighted Parameter Map; the highest absolute value in the Sum Gradient Map).
Under the default map calculation, Weighted Parameter Map, all active layers are defaulted to “all values in selected range are preferred”. This can result in large areas with equal values and subsequently poor route calculation. The example below shows a poor route calculation across a map area with a single value."
By switching the layer from “all values in selected range are preferred” to either “larger values in selected range are preferred” or “smaller values in selected range are preferred”, the route calculated is able to make a more precise calculation.