I'm working through CHAOS: An Introduction to Dynamical Systems by Alligood et. al. and I'm on Challenge 7 step 5. At this point we have a system of ODEs. (Note: a dot over a variable is its derivative with respect to time).
$$\begin{align*} \dot{x} & = z - \left(\frac{x^3}{3} - x\right) \\ \dot{z} & = -x \end{align*}.$$
We also have an energy function
$$E(x, z) = \frac{1}{2}\left(x^2+z^2\right).$$
I've been able to prove that
$$\dot{E} = \left(\frac{x^3}{3} - x\right) \dot{z}$$
using the chain rule. But, then, the book seems to be asking me to show something I can't quite get. If $X$ is a trajectory of the system of ODEs and $X_0$ a segment of it starting at $(x_1, z_1)$ and ending at $(x_2, z_2)$ then I'm supposed to show that
$$E(x_2, z_2) - E(x_1, z_1) = \int_{X_0}{\left(\frac{x^3}{3} - x\right) \dot{z}dt}.$$
Now here is my issue. I can understand if the statement is
$$E(x_2, z_2) - E(x_1, z_1) = \int_{t_1}^{t_2}{\left(\frac{x^3}{3} - x\right) \dot{z}dt}$$
where $t_1$ and $t_2$ are the times corresponding to the ends of $X_0$. But, how do I get to integrating over the segment of the trajectory as opposed to integrating over time?
Note: the book is actually asking me to prove something slightly different, but it would require a good amount of more setup and I think this amounts to the same thing.