Suppose (X, Y) is uniformly distributed over the region defined by $0 \leq y \leq \sqrt{1-x^2}$ and $-1 \leq x \leq 1$.

a. Find the marginal densities of X and Y

b. Find the conditional densities of $f_{X|Y = y}(x) and f_{Y|X = x}(y)$.

c. Are X and Y independent?

So I understand that the region is essentially a half circle and since it's a uniform distribution $f(x, y) = 1$ when $0 \leq y \leq \sqrt{1-x^2}$ and $0 \leq x \leq 1$ and $f(x, y) = 0$ otherwise.

For part A I'm a little confused on how to get the bounds for the integral to get the marginal densities, $f_X(x)$ and $f_Y(y)$. Would $f_X(x) = \int_0^\sqrt{1-x^2}dy$ and $f_Y(y) = \int_0^\sqrt{1-y^2}dx$?

For part B I know that $f_{X|Y=y}(x) = \frac{f(x,y)} {f_X(x)}$ and $f_{Y|X=x}(y) = \frac{f(x,y)} {f_Y(y)}$ If I am correct in part A, I believe $f_{X|Y=y}(x) = \frac{1}{\sqrt{1-y^2}}$ and $f_{Y|X=x}(y) = \frac{1}{\sqrt{1-x^2}}$ but I'm confused what the bounds are for this function to hold.