What does the area defined by two different values for quantity under a supply curve represent?
Your intuition is right, but you are not accounting for fixed costs.

The supply curve is the part of the marginal cost curve above its intersection with the average variable cost curve. Marginal costs are defined as the derivative of the cost function (or the variable cost function as the derivative of fixed cost is zero).

To get the area you integrate. That is, you get back the variable cost.
So the area below the supply curve between $a$ and $b$ is the difference of the variable costs.