Math / How much piles up
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Change · total
Last stop asked how steep. Flip it: you know the rate curve — rain per hour, fill per minute, speed against time. How much piles up in total? The area under that rate curve is the total. That tool is the (integral). Purpose first: how much piled up. The name comes last.
— rate. This page is total from rate.
People estimated fields and piled strips long before calculus. Archimedes exhausted circles with polygons. The job is total from pieces.
Same century as the rate tool. Leibniz’s long s (∫) is an elongated S for sum — add the thin strips. Newton’s fluxions ran the other way: from total change back to rate. Two directions, one lock.
If you pile up rates from a to b, you get the net change of the original quantity. Steepness and pile are inverses — that is the deep fact, not a slogan.
Honesty: one rate curve, one shaded total. Not a drill sheet of antiderivatives.
The curve is a rate (how much per unit run). Shade from the left up to your marker. The shaded area is the total that piled up in that window (area under rate = total).
Window to t = 3.0 · total piled ≈ 4.5
Sample rate: gentle hill of “units per hour.” Drag the right edge of the shade. Bigger window → more total (while the rate stays positive).
Integralen ∫ rate dt är arean — den totala mängden som samlats.
Same lock, two directions. You pick the direction that matches what you were given.
1. Rain. For 2 hours the rain rate is steady at 3 mm per hour. How much rain piled up?
Estimate: 3 × 2 = 6 mm — a rectangle of rate × time.
Area of rectangle: height 3, width 2 → 6 mm. Steady rate makes the “area” ordinary multiply. Changing rate needs the curved shade.
2. Link. If the rate is the slope of a tank’s level, what is the area under the rate from noon to three?
Estimate: the net rise (or fall) of the level between noon and three.
Area under rate = net change of the original quantity. Here: how much the tank level changed from noon to three.
Hands-on (drag the window) is 20. Each right gate item is 20.
1. What is this tool doing?
2. Steady rain at 4 mm/h for 3 hours. Total?
3. Area under a speed-against-time graph is…
4. Which is a real question?