  This paper investigates the exact number of limit cycles given by the
averaging theory of first order for the piecewise smooth integrable
non-Hamiltonian system \begin{eqnarray*} (\dot{x},\ \dot{y})=\begin{cases}
(-y(x+a)^2+\varepsilon f^+(x,y),\ x(x+a)^2+\varepsilon g^+(x,y)),\ \ x\geq0,\\
(-y(x+b)^2+\varepsilon f^-(x,y),\ x(x+b)^2+\varepsilon g^-(x,y)),\ ~ \, x<0,\\
\end{cases}\end{eqnarray*} where $ab\neq 0$, $0<|\varepsilon|\ll 1$, and
$f^\pm(x,y)$ and $g^\pm(x,y)$ are polynomials of degree $n$. It is proved that
the exact number of limit cycles emerging from the period annulus surrounding
the origin is linear depending on $n$ and it is at least twice the associated
estimation of smooth systems.
