Supply Function: $X_S(p)=ln(\frac{e^2}{4}p)$
Demand Function: $X_D(p)=\sqrt{20-4p}$
Equilibrium price $EP=4$ , Equilibrium quantity $EQ= 2$
$y=2$
How do you calculate the consumer/producer surplus, when the supply and demand function start at different $p$ values?
In accordance with your second image (and if I understand correctly), the consumer surplus equals the integral of the demand curve from $0$ to $EQ$ minus $EQ\cdot EP$. Similary, the producer surplus equals $EQ\cdot EP$ minus the integral from $0$ to $EQ$ of the supply curve.
Remark. You switched up the equilibirum quantity and price, since we have $X_D(4)=X_S(4)=\underbrace2_{\text{equilibrium price}}$. So we have $EQ=4$ and $EP=2$.
So the consumer surplus equals $$\int_0^{\overbrace4^{\text{EQ}}} \sqrt{20 - 4 p} \,\mathrm d p - 2\cdot 4 = \left[ -\frac14(20 - 4 p)^\frac32 \cdot \frac23 \right]^{p=4}_{p=0}-8 = \frac43(5\sqrt 5 -1)-8\approx5.57.$$
Similarly, the producer surplus equals $$8 - \int_0^4 \ln\left(\frac{\exp(2)}4 p\right)\,\mathrm dp=8-\left[p + p\log(p/4)\right]^4_0=8-4=4.$$