Given the PDF of a continuous joint distribution:$$f(x,y)$$ Why isn't it true that taking a cross sectional area of the distribution parallel to the y axis represents the following: $$\int_{-\infty}^{\infty}f(x,y)dy = P(X=x)$$
Surely integrating for all possible values of y with respect to y, is akin to summing all probabilities P(x, y), where x is fixed at a given value. If that is the case, that would be calculating the probability that the random variable X is equal to a given value x, as you have exhausted all instances where X = x. Why is it that the cross sectional area still yields a probability density as opposed to an absolute probability?
Thank you in advance.