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Use horizontal slices, right minus left, and y-coordinate bounds to represent planar area. Prerequisites: fixed-order area integrals and elementary inverse branches. Compare directions and choose one that avoids unnecessary splitting; all inversions stay elementary.
Oct 28, 2026
Find where curves exchange upper and lower positions, then sum nonnegative vertical-slice areas. Prerequisite: fixed-order area integrals. Practice distinguishes crossings, touching points and signed cancellation without adding horizontal-slicing calculations. Cover: [Area between 2 Functions](https://commons.wikimedia.org/wiki/File:Area_between_2_Functions.svg) by Yomomo, [CC BY-SA 4.0](https://creativecommons.org/licenses/by-sa/4.0/). Rasterized and padded to fit; derivative cover under the same license.
Oct 28, 2026
Set up and evaluate geometric area with vertical slices when curve order stays fixed. Prerequisites: definite-integral evaluation and elementary substitution. Use exact intersections, nonnegative slice heights and square units, including regions below the axis.
Oct 28, 2026
Compute and interpret interval averages of functions and rates, and check when continuity guarantees attainment. Prerequisites: definite integrals, integral bounds, continuity and extrema. Practice distinguishes an interval average from midpoint samples and average change. Cover: [mvt](https://commons.wikimedia.org/wiki/File:MVT.svg) by Auswahlaxiom, [CC BY-SA 4.0](https://creativecommons.org/licenses/by-sa/4.0/). Rasterized and padded to fit; derivative cover under the same license.
Oct 28, 2026
Optional proof enrichment: construct and check input tolerances for finite limits of linear and constant functions. Requires the meaning of arbitrary precision in a limit. Practice quantifier order, punctured neighborhoods and written justification.
Oct 28, 2026
Core checkpoint on average value, area, distance and feasible accumulation. Requires vertical and horizontal slicing, crossing curves, motion integrals and inflow/outflow models. Diagnose setup, units and interpretation in independent short scenarios.
Oct 28, 2026
Optional extension: model signed work by a changing collinear force and calibrate an ideal spring with consistent units. Requires definite-integral evaluation and basic substitution competence. Distinguish a force integral from an endpoint-force estimate.
Oct 28, 2026
Optional extension: construct cylindrical-shell volume models and choose a slicing method. Requires washer geometry, area slices and elementary definite integration. Practice interpreting radius as a distance, selecting thickness and comparing valid setups.
Oct 28, 2026
Optional extension: construct disc and washer integrals about horizontal or vertical axes, including shifted axes. Prerequisites: cross-sectional volumes and area between curves. Practice measuring radii as distances and matching the slicing variable to the axis.
Oct 28, 2026
Optional extension: build volume integrals from perpendicular cross-sectional areas and simple geometry. Prerequisites: FTC evaluation and elementary substitution. Practice tracking side lengths, diameters and cubic units.
Oct 28, 2026
Build stock models from inflow minus outflow, then test empty and full boundaries and interior extrema. Prerequisites: net change and accumulation functions. Practice choosing intervals where stated rates remain physically meaningful.
Oct 28, 2026
Use velocity signs and definite integrals to distinguish displacement, total distance and final position. Prerequisites: net change and definite integration. Practice identifying genuine reversals and interpreting one-dimensional motion.
Oct 28, 2026
