If f(t) is the rate of flow of a continuous income stream, then the total income produced during the period from t = a to t = b is:
Total income = ∫ab f(t) dt
If f(t) is the rate of a continuous income stream, 0 ≤ t ≤ T, and if the income is continuously reinvested at a rate r, compounded continuously, then the future value FV at the end of T years is given by:
FV = ∫0T f(t) er(T - t) dt = erT ∫0T f(t) e-rt dt
The future value of a continuous income stream is the total value of all money produced by the continuous income stream (income and interest) at the end of T years.
If (x̄, p̄) is a point on the graph of the price-demand equation p = D(x) for a particular product, then the consumers' surplus CS at a price level of p̄ is:
CS = ∫0x̄ [D(x) - p̄] dx
which is the area between p = p̄ and p = D(x) from x = 0 to x = x̄.
The consumers' surplus represents the total savings to consumers who are willing to pay more than p̄ for the product but are still able to buy the product for p̄.
If (x̄, p̄) is a point on the graph of the price-supply equation p = S(x), then the producers' surplus PS at a price level of p̄ is:
PS = ∫0x̄ [p̄ - S(x)] dx
which is the area between p = p̄ and p = S(x) from x = 0 to x = x̄.
The producers' surplus represents the total gain to producers who are willing to supply units at a lower price than p̄ but are still able to supply units at p̄.
In a free competitive market, the price of a product is determined by the relationship between supply and demand. If:
then: