AP Calculus BC / Unit 6: Integration and Accumulation of Change
6.5 Interpreting the Behavior of Accumulation Functions Involving Area
Notes
For $\displaystyle g(x)=\int_a^x f(t)\,dt$, the graph of $f$ is the graph of $g'$:
- $g$ increases where $f>0$, decreases where $f<0$.
- $g$ has extrema where $f$ changes sign.
- $g$ is concave up where $f$ is increasing.
- Values of $g$ come from areas (geometry).
Practice
$f$ is positive on $(0,3)$ and negative on $(3,5)$. $\displaystyle g(x)=\int_0^x f(t)dt$ has a local max at:
$g'=f$ changes from positive to negative.
On a graph of $f$, the area from 0 to 2 is 4 (above axis). Then $\displaystyle g(2)=\int_0^2f$ is:
The integral equals the signed area.
$g$ is concave down where:
$g''=f'$.
Ready for more? Take the Unit 6 test.