I have computed the red area.
Which I can compute with this double integral:
$$\int_{0}^{1}\int_{1}^{x+1} dy dx = 1/2$$
or with a single integral:
$$\int_{0}^{1} (x+1) dx - \int_{0}^{1} dx = 1/2$$
However, what does this integral computes, graphically?
$$\int_{0}^{1}\int_{1}^{x+1} (x+1) dy dx = 5/6$$
There is no online calculator that shows me a graph of this double integral, so I'm really struggling with this intuitively. I have come up with this problem when dealing with probabilities: I usually have an f(x,y) where I need to compute certain ranges for integrals, but I can't figure out the relationship between the domain when it's variable-dependent, and the f(x,y).