Exploring Accumulations of Change AB & BC
If $f$ is a rate of change, the area between $f$ and the horizontal axis measures the accumulated change in the underlying quantity, with the units coming from the two axes: liters per minute against minutes gives liters. Area above the axis counts positive and area below counts negative, so the accumulated change over an interval is a signed area. A flow that encloses $18$ above the axis and $9$ below leaves the tank up by $18 - 9 = 9$ liters, which is the net change.
The same picture answers a second question differently. Total area counts both pieces positively and gives $18 + 9 = 27$, the volume that actually moved, and net change and total area agree only when the rate keeps one sign. The running total is a function of where you stop: it increases wherever the rate is positive, decreases wherever the rate is negative, and is largest where the rate crosses from positive to negative, which is generally nowhere near where the rate itself is largest.
The work
3 ways in · any order
Lesson
Exploring Accumulations of Change
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Reads an amount off the area under a rate graph, gets the units by cancelling the two axes, shows the region below the axis subtracting, separates net change from total area on one picture, and tracks the running total to show it peaking where the rate crosses zero rather than where the rate is largest.
Diagnostic
10-item topic check
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Ten items spanning the two failure modes of this topic: reporting a signed net change when total area was wanted or the reverse, and reading the behaviour of the accumulated amount off the size of the rate instead of its sign.
Targeted Practice
Drill a single misconception
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Pick one of the failure modes you missed and drill it on its own. The round is adaptive: two correct in a row clears the misconception and moves you to the next.