The vertical slices of a cone form arches. A curve can be divided into many little pieces, each of which is approximately straight. Arc lengths as integrals are usually evaluated using antiderivatives. The center of mass of a mechanical system can be thought of as the balancing point for the system. The center of mass of the triangle in Example 6 is calculated to be on the x-axis.
Time has nothing to do with work.
The work done to slide a 5 lb book 3 feet across a floor is 15 foot-pounds. What goes up must come down. Water pressure is proportional to depth.