The joint probability density is $ f(x,y)$
The probability density of X from the joint density is
$$\int_{-\infty}^{\infty} \int_{A} f(x,y)dxdy$$
$$f_X(x)= \int_{-\infty}^{\infty} f(X=x,y)dy$$
$$ \implies P(X\in A)=\int_{A} f_X(x)dx$$
Can someone explain how to graphically interpret the last expression when a joint distribution is given?
Graphically, if the distribution is a massive shape described by the supports of $X,Y$, and $f_{X,Y}(x,y)$ is the shape's density of at point $(x,y)$, then $f_X(x)$ is the measure of the contribution of shape's mass by the crossection at $X=x$. "Adding"(ie integrating) these contributions in the range $X\in A$ gives the mass of that section of the shape.
Assuming that the joint, marginal, and conditional probability density functions are all well defined for the distributions of the real-valued continuous random variables $X$ and $Y$: $$\begin{split}\mathsf P(X\in A) & =\mathsf P(X\in A ~\cap~ Y\in\Bbb R) \\ &= \iint_{\Bbb R\times A} f_{\small X,Y}(x,y)~\mathsf d (y,x) \\ &= \int_A\int_\Bbb R f_{\small X,Y}(x,y)~\mathsf d y~\mathsf d x \\ &=\int_A f_{\small X}(x)\left(\int_\Bbb R f_{\small Y\mid X{=}x}(y)~\mathsf d y\right)~\mathsf d x \\ &= \int_A f_{\small X}(x)~\mathsf d x \end{split}$$
This is basically just the Law of Total Probability in action. For discrete integer valued random variables you would have the analogous: $$\begin{split}\mathsf P(X\in A) & =\mathsf P(X\in A ~\cap~ Y\in\Bbb Z) \\ &= \mathop{\sum\!\!\sum}_{(y,x)\in\Bbb Z\times A} \mathsf P(X=x,Y=y) \\ &= \sum_{x\in A} \sum_{y\in \Bbb Z} \mathsf P(X=x)~\mathsf P(Y=y\mid X=x) \\ &= \sum_{x\in A} \mathsf P(X=x)\sum_{y\in \Bbb Z}\mathsf P(Y=y\mid X=x) \\ &= \sum_{x\in A} \mathsf P(X=x) \end{split}$$