\[ M = \int_{z_{2}}^{z_{1}} t \sigma z \; dz\label{b}\tag{Eq.1} \\ F = \int_{z_{2}}^{z_{1}} t \sigma \; dz \]
Let \[ \sigma = mz + c\label{a}\tag{Eq.2} \] and \[ t = const \] where \[ m = \frac{\sigma_{1} - \sigma_{2}}{z_{1} - z_{2}} \]
\[ c = \sigma_{1} - z_{1}m \] Substitute the expression for \(\sigma\) (\(\ref{a}\)) into the expression for moment for arbitrary stress profile (\(\ref{b}\)) and hold \(t\) constant
\[\begin{align} M & = \int_{z_{2}}^{z_{1}} t (mz + c) z \; dz \\ M &= t \int_{z_{2}}^{z_{1}} mz^{2} + cz \; dz \\ M &= t \biggl[ \frac {mz^{3}}{3} + \frac{cz^{2}}{2} \biggr]_{z2}^{z1} \\ M &= t \biggl( \frac{m}{3}(z_{1}^{3} - z_{2}^{3})+\frac{c}{2}(z_{1}^{2}-z_{2}^{2})\biggr) \end{align}\]