Integral calculus
     Applications of the definite integral
      The area between two curves
         The area between two curves examples
The area between two curves
If given are two continuous functions,  and  g defined over an interval [a, b], with  g (x) <  f (x)  for all x in [a, b], then the area A of the region bounded (or enclosed) by these two curves and the lines x = a and x = b is given by
 
The area between two curves examples
Example:   Find the area of the region bounded by the curve  f (x) = - x2 - 2x and the line  g(x) = x.
Solution:   To find points of intersections (limits of integration) of the given functions we solve their equations,
  - x2 - 2x = x   or    x2 + 3x = 0, 
x(x + 3) = 0,    x1 = 0 and  x2 = -3.
Thus, the area
 
Example:   Find the area of the region lying to the left of the y-axis and enclosed by the curves,
 f (x) = (1/3)x3 - 3x and  g(x) = x2 + 3x.
Solution:   To find points of intersections (limits of integration) of the given curves we solve their equations,
 
Thus, the area
Example:   Find the area of the region bounded by the curve  y2 = x and the line  y = - x + 2.
Solution:   To find points of intersections (limits of integration) of the given curve and the line we solve their 
equations,    (- x + 2)2 = x   or   x2 - 5x + 4 = 0, 
(x - 1)(x - 4) = 0,    x1 = 1  and  x2 = 4.
Thus, the area
 
The same area can be calculated by changing the role of the variables (or the coordinate axes) to get simpler integral expression, therefore
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