# FV Discretization of source term in 2D Poisson Equation

I am learning Finite Volume method using the textbook "An Introduction to CFD: Finite Volume Method" by Veersteg and Malalasekera.

I would like to solve a heat conduction problem over a square grid $$0.5m$$ in length with all the boundaries maintained at $$0°C$$. For that, I need to discretize the Poisson equation:

$$\frac{d^2 T}{d x^2} + \frac{d^2 T}{d y^2} = 32(x(x-1) + y(y-1))$$

But I am confused about how to discretize the source term in this since the book does not deal with 2-D non-constant source term. My guess is that we separately integrate each of the terms by $$dxdy$$ as below:

$$\left (A_e \frac{\partial T}{\partial x} - A_w \frac{\partial T}{\partial x}\right) + \left (A_n \frac{\partial T}{\partial y} - A_s \frac{\partial T}{\partial y}\right) = 32x^2dxdy -32xdxdy + 32y^2dxdy - 32ydxdy$$

which becomes:

$$\left(A_e \frac{T_e - T_p}{\delta x_{ep}} - A_w \frac{T_p - T_w}{\delta x_{pw}} \right) + \left(A_n \frac{T_n - T_p}{\delta y_{np}} - A_s \frac{T_p - T_s}{\delta y_{ps}} \right) = \frac{32}{3}(x^3y + y^3x) - 16(x^2y + y^2x)$$

But I am not sure if this is the right wat to go about it.

I tried to find resources on the internet about this but all I could find was a few academic papers that were not easy to understand.

So what is the correct way to discretize the Poisson equation?