Исследование накопления изменений

Adding up change to get a total

• The derivative took things apart into rates. Integration puts them back together into totals.
• If you know the rate something changes, you can find the net change 净变化 it accumulates.
• Drive at a known speed for a while → total distance is speed accumulated over time.
• This "adding up a rate" is the second big idea of calculus: accumulation 累积.

Area under a rate graph = accumulated amount

• Plot a rate of change against time. The area 面积 under that graph is the total accumulated change.
• Constant rate $60\,\tfrac{\text{km}}{\text{h}}$ for $2$ h → area of a rectangle $=60\times2=120$ km.
• A changing rate makes a curvy region, but the idea is the same: area = accumulated quantity.
• Height is the rate; width is the interval; area is the total.

The area under a rate graph is the total amount accumulated

Signed area: rates can be negative

• When the rate is positive, the quantity grows (area counts as positive).
• When the rate is negative (below the axis), the quantity shrinks (area counts as negative).
• The net change is signed area 有符号面积: positive parts minus negative parts.
• So an object moving backward subtracts from the accumulated displacement.

From rate back to amount

• If $f$ is a rate and you accumulate from $a$ to $b$, you get the net change of the original quantity.
• This is exactly the reverse of differentiating — undoing the rate to recover the total.
• The tool that computes this signed area precisely is the definite integral (coming up).
• For now: area under a rate graph = the amount that accumulated.