The definite integrals section has defined that the area under a curve of function y = f(x) between x = a and x = b equals the definite integral

In this section, integrals are used to find areas between curves.

Consider the region S, shown on the left, lies between two curves y = f(x) and y = g(x), and between two vertical lines x = a and x =b, where f and g are continuous functions and f(x)g(x) for all the x in [a, b].

To find the area of S, divide the interval [a, b] into n pieces. The width of each piece is

      Δx = (b-a)/n

The left endpoint for a given sub-interval i is

      x_i = a + idx

in which i = 0, 1, 2, ..., n-1.

Consider a typical rectangle (red in the left diagram) between x_i and x_i+1, Choose a x in [x_i, x_i+1], say x_i^*, then the height of this typical rectangle is

      h = f(x_i^*) - g(x_i^*)

The width of this typical rectangle is

      w = x_i+1 - x_i = Δx

Then the area of this typical rectangle is

      ΔA = (f(x_i^*) - g(x_i^*))Δx

The Rieman sum gives the approximate area of S.

Since f and g are continuous, the limit of this Rieman sum exits. Therefore, the area of the region bounded by the curves y = f(x) , y = g(x), and the lines x = a and x = b, where f and g are continuous and and f(x ) g(x) for all the x in [a, b], is

Area between Curve: f and g are Positive

Area between Curve: S is split into
Small Region

If both f and g are positive, the area between these two curves can be find out directly from the definite integral.

       A =[area under y = f(x)] - [area under y =g(x)]
      

If f(x ) g(x) for some values of x and g(x ) f(x) for some values of x in [a, b], then region S is split into several regions S_1, S₂, ... with area A_1, A₂, .... The area A of region S equals the sum of area A_1, A₂, ....

      A = A₁+ A₂ + ...

Since
      

The area between the curves y = f(x) and y = g(x) and between x =a and x =b is

In some cases, the area between curves is easier to calculate if the function are in terms of y, instead of x. If a region S is bounded by curves x = f(y) and x = g(y), and lines y = c and y = d, where f and g are continuous, the area of S is