Definite Integrals


   
 
Applications of Definite Integrals
Let y = f (x) be a curve. The area bounded by y = f (x), x-axis and the ordinates at x = a and x = b is given by
 
 
 
(ii) The area bounded by the curve x = f (y)
 
y = axis and the abscissae at y = c and y = d is given by
 
 
 
Note 1: The area bounded by the curves f(x) and g(x) and the ordinates x = a and x = b is given by
 
 
 
Note 2: If curve f (x) lies above the x-axis and g (x) lies below the x-axis then area bounded by f (x) and g (x) is
 
 
 
Note 3: If f (a) and f (b) are opposite in sign then the curve crosses the x-axis say at c.
 
Then the area bounded the curve f (x), x-axis and the ordinates x = a and x = b is
 
 
 
Example:
 
Find the area of the region between the X-axis and the graph of f(x) = x3 - x2 - 2x   -1 x < 2.
 
Solution:
 
Let us find the values of x in the given interval, for which
 
f(x) = 0
 
x3 - x2 - 2x = 0
 
x(x2 - x - 2) = 0
 
 
 
Partition the domain [-1, 2] to subintervals [-1, 0] and [0, 2]
 
 
 
 
 
 
 
 
 
 
     
   
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