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In the present paper, we study the number of zeros of the first orderMelnikov function for piecewise smooth polynomial differential system, toestimate the number of limit cycles bifurcated from the period annulus ofquadratic isochronous centers, when they are perturbed inside the class of allpiecewise smooth polynomial differential systems of degree $n$ with thestraight line of discontinuity $x=0$. An explicit and fairly accurate upperbound for the number of zeros of the first order Melnikov functions withrespect to quadratic isochronous centers $S_1, S_2$ and $S_3$ is provided. Forquadratic isochronous center $S_4$, we give a rough estimate for the number ofzeros of the first order Melnikov function due to its complexity. Furthermore,we improve the upper bound associated with $S_4$, from $14n+11$ in \cite{LLLZ},$12n-1$ in \cite{SZ} to $[(5n-5)/2]$, when it is perturbed inside all smoothpolynomial differential systems of degree $n$. Besides, some evidence on theequivalence of the first order Melnikov function and the first order Averagedfunction for piecewise smooth polynomial differential systems is found.

Limit cycles by perturbing quadratic isochronous centers inside piecewise smooth polynomial differential systems