Find the position function from the piecewise-defined velocity function

Question

I am getting stuck on a position function problem in my Diff Eq class. Problem 22 is shown on the right in the picture below. On the left is the answer. My work below shows that I get stuck calculating the second interval 3 geq t leq 7. It looks like the initial position is supposed to be negative and I calculate it as positive?

Answer 1

Here is a hint to get you started: Since $x(0)=0$,

$x(t)=x(t)-x(0)=\int_0^t v(t)dt\;\;$ where $v(t)=\begin{cases}\frac{5}{3}t &\mbox{, if } 0\le t\le3\\5 &\mbox{, if }3\le t\le 7\\-\frac{5}{3}t+\frac{50}{3} &\mbox{, if } 7\le t\le10\end{cases}$