Applications: area between two curves and mean value

Area between two curves

If f and g are continuous on [a;b] and if f(x) ≥ g(x) for all x in [a;b], the area of the region between the two curves is:

A = ∫_(a)^(b) (f(x) − g(x)) dx (in units of area)

Example: f(x) = 6 − x and g(x) = x². We solve f(x) = g(x): 6 − x = x², which gives x² + x − 6 = 0. This can be factored as (x − 2)(x + 3) = 0, so x = 2 or x = −3. On [-3;2], f(x) ≥ g(x). The area is the integral from -3 to 2 of (6 − x − x²) dx.

Mean value of a function

The mean value of f on [a;b] is defined by:

m = (1/(b-a)) * integral from a to b of f(x) dx

Example: mean value of f(x) = x² on [0;3]: m = (1/3) * ∫₀³ x² dx = (1/3) * (27/3) = (1/3) * 9 = 3.

Common pitfall

When calculating the area between two curves, one must always check the sign of f(x) – g(x) over the entire interval under consideration. If the curves intersect several times between a and b, the interval must be divided at each point of intersection before integrating; otherwise, the positive and negative areas will be incorrectly cancelled out.