AP Calculus AB and BC glossary

Accumulated change

Also called: Accumulation of change

Accumulated change is the net change in a quantity over an interval, obtained by integrating that quantity's rate of change across the interval. It carries the units of the rate multiplied by the units of the variable integrated over, and it is a net figure, so falls cancel earlier rises.

accumulated change=∫abr(t) dt\text{accumulated change} = \int_{a}^{b} r(t)\,dt

Read it off the graph of the rate itself. The accumulated change between t=at = a and t=bt = b is the area between the curve r(t)r(t) and the horizontal axis, counted positively where the curve sits above the axis and negatively where it sits below. The axis labels then hand you the units: a vertical axis in people per year against a horizontal axis in years gives an answer in people, since the integral multiplies heights in rr by widths in tt.

The word net carries weight. Wherever r(t)r(t) is negative the quantity falls, and that fall subtracts from what earlier growth contributed, so an accumulated change of zero can hide a great deal of movement. A particle whose velocity changes sign accumulates a displacement smaller than the distance it travelled, and recovering distance takes ∫ab∣v(t)∣ dt\int_a^b \left|v(t)\right|\,dt instead.

The mistake

Reporting the integral as the amount present. The value of ∫05r(t) dt\int_0^5 r(t)\,dt is how much the quantity changed over those five units of time, not how much there is at t=5t = 5. When the question wants the amount rather than the change, reach for the net change theorem, which carries the form that starts from the amount already there.

Appears in: Unit 6: Integration and Accumulation, Unit 8: Applications of Integration