Definite integrals, areas and volumes
Definite integrals, areas and volumes · Diagrams
Derive upper minus lower
Build area from upper minus lower
Illustrative curves f(x) = 2x − x² and g(x) = x² on [0, 1]. Inspect a strip's two heights before multiplying their difference by its width.
Teaching and source notes
The source derives A = F − G by subtracting the area under the lower curve from that under the upper curve, between their meeting points a and b. The numerical formulas here are labelled illustrative rather than attributed to the original sketch.
For f(x) = 2x − x² and g(x) = x², f − g = 2x(1 − x) is nonnegative on [0, 1]. The curves meet at both endpoints. Integrating the upper curve gives 2/3, the lower gives 1/3, and their difference is 1/3.
The same bounds and consistent upper/lower ordering are essential. If curves cross within an interval, split before summing geometric areas. A finite strip sum is still an approximation.