Calculating the space a complex shape takes is hard, but it's not hard to calculate the space of a rectangle, so why not use smaller and smaller rectangles.

Join me as I re-invent integrals without realizing what I am doing.

@olafurw This sounds more like the pre-calculus geometric appeal to infinitesimals. That they were able to produce answers in a simpler way than classic geometry was scandalous to some. Newton avoided geometric infinitesimals by inventing fluxions and thinking of things moving and time was divided into infinitesimals. Much less upsetting to the math conservatives.