1. The problem involves integrating with respect to $z$ from $0$ to $\frac{L}{2}$, not $t$ or $1$. 2. We need to set up the integral correctly with the limits $0$ to $\frac{L}{2}$. 3. The integral will be of the form $$\int_0^{\frac{L}{2}} f(z) \, dz$$ where $f(z)$ is the function to integrate. 4. Make sure to substitute the correct limits in the integral evaluation step. 5. If the function $f(z)$ is known, perform the integration and then evaluate at $z=\frac{L}{2}$ and $z=0$. 6. The definite integral value is $$F\left(\frac{L}{2}\right) - F(0)$$ where $F(z)$ is the antiderivative of $f(z)$. 7. This correction ensures the integral is computed over the correct variable and limits as specified.