Expectation as Integral: Visualizing the area under the curve For continuous variables we can define expectation as 
$E(X) = \int_\Omega X \, dP$
hopefully where dP = p(x) dx and $\Omega$ being the range in which X can fall.
Area under which curve are we trying to find? What's on the x-axis, what's on the y-axis and why is this area giving expectation?
Let's consider a simple example: X is a random variable in range [0,1] and all values between 0-1 (continuous) are equally likely, p(X) is simply x-axis, isn't it? And this makes it extremely hard to visualize the integrand $x \, p(x)$ of 
$E(X) = \int_\Omega x \, p(x) \, dx$
unlike 

where the integrand f(x) is straightforward to imagine and it is easy to visualize what we are trying to evaluate as above. Help me visualize and understand what we are trying to evaluate here.
 A: An integration, is a product of variables. Distance traveled is the product of velocity and time. In the case of expected value it could be n times p, where p varies with time. In that case, the number of trials times the probability of a certain outcome is not a simple product. In fitting this to a function where the definite integral would be the expected value...... 
Example: What is the expected number of defective items over a one month period as a manufacturer implements a continuous quality control improvement plan whereby the probability of a defect is some function f(t). This could be a decreasing number of defective items over time (production rate times decreasing probability). The definite integral evaluated from t1 to t2 would be the expected number of defective items. 
A: The set of probabilities $\mathbb{P}(X\leq x)$, $x\in \mathbb{R}$ is the distribution function of $X$, which we usually denote as $F_X$. A continuous distribution function is of the form
$$F_X (x)=\int_{-\infty}^{x}f_X(x)dx$$
where $f_X(x)$ is called the density function and it is such that $f_X(x) \geq 0 \quad \forall x$, $\int_{-\infty}^{\infty}f_X(x)dx=1$ and $\int_{a}^{b}f_X(x)dx=\mathbb{P}(a \leq X\leq b)$.
In order to visualize the area under the curve, draw the values of $f_X(x)$ on the y-axis (vertical axis) and the values of $x \in X$ on the horizontal axis. As the integral is the limit of a sum of rectangles, the area under the curve helps you visualize $\mathbb{P}(a \leq X\leq b)$.
Notice that the above expressions are different than $\mathbb{E}(X)=\int_{-\infty}^{\infty}Xf_X(x)dx$, which is a value on the horizantal axis.
