Continuous function supported in $[2-\delta,2]$, satisfying integral of $f(x)\sqrt{4-x^2}$ equals $1$.

Fix $\delta >0$.I want to show the existence of continuous function $f$, satisfying $supp(f) \subset [2-\delta, 2]$ and $$\frac1{2\pi}\int_{2-\delta}^{2} f(x)\sqrt{4-x^2}\,\mathrm dx=1\;.$$

How do I prove this? Will mollifier be used?