An Intuitive Way to Define Triple Integrals
A visual guide to turning 3D solids into integration bounds.
The process of solving triple integrals is generally quite mechanical and not very difficult to learn. The real trouble tends to arise before the actual integration takes place: determining the limits, which usually involves expressing a three-dimensional shape in mathematical terms.
This project was intended to provide learners with an intuitive method for tackling these problems by teaching them how to move between dimensions.
Rather than considering all six boundaries at once, the student begins in three dimensions and concentrates on the direction of the first variable. They then determine both where they enter and where they leave the solid. After grasping these boundaries, they project the solid into two dimensions, in which the other boundaries become far easier to describe.
Two examples are carefully gone through step by step, with the reasoning clearly shown at each stage. The learner is then given a new problem and is guided through it by means of questions in order that they may apply the same method with gradually decreasing support.