SZ effect is given by $${\Delta T_\rmn{SZ} \over T_\rmn{CMB}} = f(x)y,$$ where $x$ is a dimensionless frequency equal to $h\nu/k_\rmn{B}T_\rmn{CMB}$ and $y$ is the comptonisation parameter, which is equal to the integrated optical depth multiplied by the fractional gain in energy per scattering. Hence $y$ is proportional to the integral of $n_\rmn{e}T_\rmn{e}$ along the line of sight $$y = \int n_\rmn{e}{k_\rmn{B}T_\rmn{e}\over
    m_\rmn{e}c^2}\sigma_\rmn{T}\rmn{d}l,$$ The frequency dependency $f(x)$ also has a dependence on the electron temperature $T_\rmn{e}$, given by $$f(x, T_\rmn{e}) = \left(x{\rmn{e}^{x}+1\over \rmn{e}^{x}-1} - 4\right)(1 +
  \Delta(x, T_\rmn{e})),$$ where the relativistic correction $\Delta(x, T_\rmn{e})$ is derived from higher-order corrections to the Kompaneets equation [@Challinor:1998; @Itoh:1998].

In the general case of modelling a non-isothermal cluster the product of pressure and $f(x, T_\rmn{e})$ is integrated along the line-of-sight. Replacing $n_\rmn{e}$ and $T_\rmn{e}$ by the radial dependence of the electron pressure, the expression for the change in the CMB temperature is given by $$\label{equation:sz_temp}
  \Delta T_\rmn{SZ} = T_\rmn{CMB}\int f(x, T_\rmn{e}) {P_\rmn{e}\sigma_\rmn{T}\over
    m_\rmn{e}c^2}\rmn{d}l,$$ which is then solved numerically.

The integral of pressure over the volume of the cluster is a measure of the total thermal energy and is therefore a useful quantity in characterising the overall thermal SZ effect. The integral of $y$ over the 