$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\Aex}{{A_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\x}{\boldsymbol{x}}
\newcommand{\X}{\boldsymbol{X}}
\renewcommand{\u}{\boldsymbol{u}}
\renewcommand{\v}{\boldsymbol{v}}
\newcommand{\e}{\boldsymbol{e}}
\newcommand{\f}{\boldsymbol{f}}
\newcommand{\dfc}{\alpha} % diffusion coefficient
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\Iz}{\mathcal{I}_z}
\newcommand{\If}{\mathcal{I}_s} % for FEM
\newcommand{\Ifd}{{I_d}} % for FEM
\newcommand{\Ifb}{{I_b}} % for FEM
\newcommand{\sequencei}[1]{\left\{ {#1}_i \right\}_{i\in\If}}
\newcommand{\basphi}{\varphi}
\newcommand{\baspsi}{\psi}
\newcommand{\refphi}{\tilde\basphi}
\newcommand{\psib}{\boldsymbol{\psi}}
\newcommand{\sinL}[1]{\sin\left((#1+1)\pi\frac{x}{L}\right)}
\newcommand{\xno}[1]{x_{#1}}
\newcommand{\Xno}[1]{X_{(#1)}}
\newcommand{\xdno}[1]{\boldsymbol{x}_{#1}}
\newcommand{\dX}{\, \mathrm{d}X}
\newcommand{\dx}{\, \mathrm{d}x}
\newcommand{\ds}{\, \mathrm{d}s}
$$
Integration over a reference element; element vector
$$
\begin{align}
\tilde b^{(e)}_{0}
&= \int_{-1}^1 f(x(X))\refphi_0(X)\frac{h}{2} dX\nonumber\\
&= \int_{-1}^1 (x_m + \half hX)(1-(x_m + \half hX))
\half(1-X)\frac{h}{2} dX \nonumber\\
&= - \frac{1}{24} h^{3} + \frac{1}{6} h^{2} x_{m} - \frac{1}{12} h^{2} - \half h x_{m}^{2} + \half h x_{m}
\tag{9}\\
\tilde b^{(e)}_{1}
&= \int_{-1}^1 f(x(X))\refphi_1(X)\frac{h}{2} dX\nonumber\\
&= \int_{-1}^1 (x_m + \half hX)(1-(x_m + \half hX))
\half(1+X)\frac{h}{2} dX \nonumber\\
&= - \frac{1}{24} h^{3} - \frac{1}{6} h^{2} x_{m} + \frac{1}{12} h^{2} -
\half h x_{m}^{2} + \half h x_{m}
\end{align}
$$
\( x_m \): element midpoint.
