# 2.11 The Finite Element Method for Two-Dimensional Diffusion

## 2.11.5 Integration in the Reference Element

The reference element can also be used to evaluate integrals. For example, consider the evaluation of the forcing function integral within an element:

 $\int _{\delta \Omega _ k} w(\vec{x})\, f(\vec{x})\, dA.$ (2.285)

In transforming the integral from $$(x,y)$$ to $$(\xi _1,\xi _2)$$, the differential area of integration must be transformed using the following result:

 $dA = dx\, dy = J d\xi _1\, d\xi _2 = J\, dA_{\xi }. \label{equ:Ax_ to_ Axi}$ (2.286)

Thus, the integrals can now be evaluated in reference element space,

 $\int _{\Omega _{\xi }} w(\vec{x}(\vec{\xi }))\, f(\vec{x}(\vec{\xi }))\, J dA_{\xi }.$ (2.287)