Properties of Definite Integrals and Accumulation Functions from a Graph

Algebra of integrals

Constants factor out and sums split, but there is no product rule: $\int fg\neq\int f\cdot\int g$. For even $f$, $\displaystyle\int_{-a}^{a}f=2\int_0^a f$; for odd $f$ it is $0$. Given $\int_0^3 f=5$ and $\int_0^7 f=2$, splitting gives $\int_3^7 f=-3$.

Accumulation functions from a graph

The signature AB free-response: the graph of $f$ is drawn from line segments and semicircles, and $g(x)=\displaystyle\int_{a}^{x}f(t)\,dt$. Then:

• $g(x)$ is the signed area from $a$ to $x$ — triangles $\frac12bh$, semicircles $\frac{\pi r^2}{2}$, negative below the axis; and $g(a)=0$.
• $g'(x)=f(x)$: $g$ increases where $f\gt0$, and $g$ has a local maximum where $f$ changes from positive to negative.
• $g''(x)=f'(x)$ is the slope of the drawn graph: $g$ is concave up where $f$ is increasing, and $g$ has an inflection point where $f$ has a local extremum.

Worked example: $f(t)=2t$ on $[0,3]$, then $f(t)=6$ on $[3,5]$, with $g(x)=\displaystyle\int_0^x f(t)\,dt$. Then $g(3)=\frac12\cdot3\cdot6=9$, $g(5)=9+6\cdot2=21$, $g'(4)=f(4)=6$, and $g''(2)=f'(2)=2\gt0$, so $g$ is concave up at $x=2$. Every answer is read off the picture.