Exploring Accumulations of Change
Introduced| English |
|---|
| net change/net tʃeɪndʒ/ |
| accumulation/əˌkjuːmjʊˈleɪʃn/ |
| area/ˈeərɪə/ |
| signed area/saɪnd ˈeərɪə/ |
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.
Integration (accumulation) is essentially the reverse of...
Accumulating a rate undoes differentiating a total.
The total change accumulated from a rate over an interval is called the ____ change.
Net change = signed accumulation.
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.
Area under a rate is a total
y = 3 (constant rate)
The shaded area under a rate curve is the accumulated amount — widen the interval and the total grows.
A tap runs at $3$ L/min for $4$ min. The area under the rate graph (accumulated volume) is...
Rectangle area $=3\times4=12$ L.
A rate rises linearly from $0$ to $6$ over $4$ min (a triangle). The accumulated amount (area) is...
Triangle area $=\tfrac12\times4\times6=12$.
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 带符号面积: positive parts minus negative parts.
- So an object moving backward subtracts from the accumulated displacement.
Where a rate graph dips below the axis, that area subtracts from the net accumulated change.
Below-axis area is negative (signed area).
On a velocity-time graph, a portion below the axis means the object...
Negative velocity = backward motion = negative signed area.
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.
Accumulated change is signed area, not just "area." A velocity graph that dips below the axis means the object moved backward, which subtracts from the net displacement. Don't add all the area as positive — regions below the axis count as negative.
A tap fills a tank at a rate of $3\,\tfrac{\text{L}}{\text{min}}$ for $4$ minutes, then $0$ after.
- The rate graph is a rectangle: height $3$, width $4$.
- Accumulated volume = area $=3\times4=12$ L.
- If the rate later went negative (draining), that area would subtract from the total.
Accumulation turns a rate back into a total: the signed area under a rate-of-change graph over $[a,b]$ is the net change of the quantity. Positive rate → area adds; negative rate → area subtracts. The definite integral makes this precise.