Exploring Volumes Bounded by Implicit Surfaces: A Visual and Multivariate Calculus-Based Approach

A discussion of the volumes of an assortment of 3D shapes defined by implicit surfaces, using both analytical and numerical methods. We begin with foundational concepts in Multivariable Calculus including triple integrals, Jacobians, etc, laying the groundwork for exact volumes of classical shapes, such as spheres, ellipsoids and torus. For more complex surfaces, Monte Carlo estimation, an useful method for estimating volumes, is employed. We then explore parameter variations to demonstrate the efficiency and versatility of combining exact integration with computational methods.