Originally Posted by
SlipEternal There are many integrals for f(x). Infinitely many, in fact. You need to use an initial condition to determine the correct constant of integration. You have:
$\displaystyle f(x) = \begin{cases}3 & 0\le x \le 3 \\ x & 3 \le x \le 5 \\ 5 & 5 \le x \le 7\end{cases}$
So, the integral of f on [5,7] is:
$\displaystyle A(x) = 5x+C$
On the interval [3,5], you have:
$\displaystyle A(5) = \dfrac{5^2}{2}+\dfrac{9}{2} = 17$
So, now, you want $\displaystyle A(5) = 5(5)+C = 17 \Longrightarrow C = -8$.
Basically, at each point between two intervals, you want the A(x) function to be continuous, so you find the value of the constant of integration to make it continuous.