$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\Aex}{{A_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\x}{\boldsymbol{x}}
\newcommand{\X}{\boldsymbol{X}}
\renewcommand{\u}{\boldsymbol{u}}
\renewcommand{\v}{\boldsymbol{v}}
\newcommand{\e}{\boldsymbol{e}}
\newcommand{\f}{\boldsymbol{f}}
\newcommand{\dfc}{\alpha} % diffusion coefficient
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\Iz}{\mathcal{I}_z}
\newcommand{\If}{\mathcal{I}_s} % for FEM
\newcommand{\Ifd}{{I_d}} % for FEM
\newcommand{\Ifb}{{I_b}} % for FEM
\newcommand{\sequencei}[1]{\left\{ {#1}_i \right\}_{i\in\If}}
\newcommand{\basphi}{\varphi}
\newcommand{\baspsi}{\psi}
\newcommand{\refphi}{\tilde\basphi}
\newcommand{\psib}{\boldsymbol{\psi}}
\newcommand{\sinL}[1]{\sin\left((#1+1)\pi\frac{x}{L}\right)}
\newcommand{\xno}[1]{x_{#1}}
\newcommand{\Xno}[1]{X_{(#1)}}
\newcommand{\xdno}[1]{\boldsymbol{x}_{#1}}
\newcommand{\dX}{\, \mathrm{d}X}
\newcommand{\dx}{\, \mathrm{d}x}
\newcommand{\ds}{\, \mathrm{d}s}
$$
Integral transformation
Reference element integration: just change integration variable
from \( x \) to \( X \). Introduce local basis function
$$
\refphi_r(X) = \basphi_{q(e,r)}(x(X))
$$
$$
\tilde A^{(e)}_{r,s} = \int_{\Omega^{(e)}}\basphi_{q(e,r)}(x)\basphi_{q(e,s)}(x)dx
= \int\limits_{-1}^1 \refphi_r(X)\refphi_s(X)\underbrace{\frac{dx}{dX}}_{\det J = h/2}dX
= \int\limits_{-1}^1 \refphi_r(X)\refphi_s(X)\det J\,dX
$$
$$
\tilde b^{(e)}_{r} = \int_{\Omega^{(e)}}f(x)\basphi_{q(e,r)}(x)dx
= \int\limits_{-1}^1 f(x(X))\refphi_r(X)\det J\,dX
$$
