I am looking the following:
The pair of random variables $(X,Y)$ is uniformly distributed on disc of radius $R$ and center $(0,0)$. Let $Z=X^2+Y^2$. I want to find the density $f_Z(t)$ for small $t$.
For that do we use that the intergral of density has to be equal to $1$ to calculate $f_z(t)$ ?
Or is tere an other way?
On
The $\text{cdf}$ is such that
$$\text{cdf}_Z(t)=\mathbb P(Z\le t)=\mathbb P(X^2+Y^2\le t)=\frac{\pi t}{\pi R^2}$$
which is simply the ratio of the areas, as the distribution is uniform.
The $\text{pdf}$ follows by differentiation on $t$,
$$\text{pdf}_Z(t)=\frac{1}{R^2}.$$
Note that $t$ small means $0\le t\le R$. Elsewhere, the $\text{pdf}$ vanishes.
Note that $Z \leq t$ iff $(X,Y)$ lies inside the circle of radius $\sqrt t$ centered at the origin.
$P(Z \leq t)=\frac {\pi t} {\pi R^{2}}$ since the area of the circle of radius $\sqrt t$ around the origin is $\pi t$. Hence $f_Z(t)=\frac 1 {R^{2}}$ for $0 <t < R^{2}$.