AP Calculus AB / Unit 8: Applications of Integration
8.3 Using Accumulation Functions and Definite Integrals in Applied Contexts
Notes
Amount at time $b$ = amount at time $a$ + $\displaystyle \int_a^b$ (rate in − rate out) $dt$.
- The integral of a rate is the total change in the quantity.
- Units: (rate units) × (time units).
Practice
Water enters at $R(t)$ gallons/hour. $\displaystyle \int_0^3 R(t)\,dt$ means:
Integral of a rate gives the total change.
A tank has 50 gallons at $t=0$. In: $\displaystyle \int_0^4 R=30$. Out: $\displaystyle \int_0^4 D=12$. Gallons at $t=4$:
$50+30-12$.
The units of $\displaystyle \int_0^{10}r(t)\,dt$ with $r$ in people/day and $t$ in days are:
People per day times days.
Ready for more? Take the Unit 8 test.