Chapter 5.3, Problem 17E

Area functions for the same linear function Let f(t) = t and consider the two area functions $A(x)={\displaystyle {\int }_0^{x}f(t)dt}$ and $F(x)={\displaystyle {\int }_2^{x}f(t)dt}$.

1. a. Evaluate A(2) and A(4). Then use geometry to find an expression for A(x), for x ≥ 0.
2. b. Evaluate F(4) and F(6). Then use geometry to find an expression for F(x), for x ≥ 2.
3. c. Show that A(x) − F(x) is a constant and that A′(x) = F′(x) = f(x).