Unit 11: From a rate to a total, and the shortcut
About this unit
An integral adds up a rate over an interval, counting time below the axis as negative, and it equals the change F(b) − F(a) in any function whose derivative is that rate.
By the end, you can: See an integral as the limit of slice-and-sum totals of a rate, with parts below the axis counting negative, and compute it exactly as F(b) − F(a) for any F with F′ = f.
Modules
5 modulesLearn
- Adding up a climb rateArticle · Slice time, treat the rate as steady on each slice, add rate × width, and let slices shrink: the integral. Negative rates give negative rectangles, so the integral is a net change.Not started
- The shortcut: F(b) − F(a)Article · The running total grows at the rate, so it shares its derivative with any antiderivative F; they differ by a constant, which cancels: ∫ₐᵇ f = F(b) − F(a).Not started
- Slice until it settlesExperiment · Add rectangles until the left sum is within 0.25 m of the exact 3 m, and watch the running total peak where the rate crosses zero.Not started
- Integrals in briefReview · Slice and sum, signed area, and the fundamental theorem in one place.Not started