Many standard problems in calculus can be easily solved by an innovative
visual approach that makes no use of formulas. The method is based on Mamikon’s
sweeping-tangent theorem, a geometrically intuitive result that is easily understood by
very young students. In this paper, the method of sweeping tangents introduced and
shown how it can be used to find (without the machinery of calculus) areas of many
plane regions, including an oval ring, a parabolic segment, a hyperbolic segment, the
region under a general power function, an exponential curve, a logarithmic curve, a
tractrix, the region between two curves traced by the rear and front wheels of a bicycle,
the region enclosed by a cardioid, and by each member of a family of lima¸cons.
The treatment of the parabolic segment and the exponential use geometric properties
of subtangents to these curves, which can also be used to draw tangent lines. An unexpected
application of the method of sweeping tangents is to physics. In this application,
conservation of angular momentum in a central force field is deduced as an elementary
consequence of Mamikon’s sweeping-tangent theorem.