Regular readers know that I am a big fan of Better Explained in which Kalid makes mathematical ideas accessible.
Integration is just multiplication when one of the operands is changing.
Most people grok integration as area under a curve but, as Kalid explains, area is just one convenient way of visualizing multiplication…but we don’t need to visualize multiplication as multiplication is already pretty straightforward – in the simplest case, it’s just repeated addition.
Many ideas in maths start out simple like that and then gradually generalize to a more complex idea. In Kalid’s words:
Our understanding of multiplication changed over time:
- With integers (3 Ã— 4), multiplication is repeated addition
- With real numbers (3.12 x sqrt(2)), multiplication is scaling
- With negative numbers (-2.3 * 4.3), multiplication is flipping and scaling
- With complex numbers (3 * 3i), multiplication is rotating and scaling
Weâ€™re evolving towards a general notion of â€œapplyingâ€ one number to another, and the properties we apply (repeated counting, scaling, flipping or rotating) can vary. Integration is another step along this path.
In other words,
Integration is just a better multiplication
Multiplication is a special case of integration when the values are static.