8.3 Accumulation in Applied Contexts

Totals from rates

Integrate a rate over time to get an accumulated total. If water enters at rate r(t), the amount added over [0, 5] is ∫[0 to 5] r(t) dt.

Net change

Net change = ∫ (rate in − rate out) dt, and final amount = initial amount + ∫ net rate dt. The result's units are the quantity's units (e.g. gallons).

Key terms

Accumulated total

Integrate a rate over time.

Amount added

∫ r(t) dt for inflow rate r.

Net change

∫ (rate in − rate out) dt.

Final amount

Initial + ∫ net rate dt.

Units

Rate × time gives the quantity's units.

Try an AP-style question

A tank holds 50 L at t = 0, and water enters at r(t) L/min. Which expression gives the amount at t = 10?

1. ∫₀¹⁰ r(t) dt
2. 50 + r(10)
3. 50 + ∫₀¹⁰ r(t) dt
4. 50 · r(10)