Visual calculus

Visual calculus, developed by Mamikon Mnatsakanian, offers a unique approach to solving integral calculus problems with minimal computation. The method was first applied in 1959 to determine the area of an annulus using tangents rather than traditional algebraic methods. By translating these tangents, a circular disk is formed, simplifying area calculations. Mamikon's theorem states that the area of a tangent sweep equals its cluster, regardless of the original curve's shape.

Mamikon collaborated with Tom Apostol on "New Horizons in Geometry," detailing his method. The approach avoids complex algebra by visualizing tangents' movement and their resulting shapes. For example, calculating the area of a cycloid involves comparing it to an enclosing rectangle and the generating circle. The theorem shows the cycloid's area is three times that of the generating circle.

This method simplifies problems like finding annulus areas and extends to various geometric shapes, demonstrating the power of visual reasoning in calculus.