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DEFINITE INTEGRATION

2.1      GEOMETRICAL INTERPRETATION OF DEFINITE INTEGRAL

If f(x) > 0 for all x ∈ [a, b]; then ∫ba f(x) is numerically equal to the area bounded by the curve y = f(x), then x-axis and the straight lines x = a and x = b i.e.  ∫ba f(x)

 

                                1251_algebraic sum.JPG

 

In general ∫ba f(x) dx represents to algebraic sum of the figures bounded by the curve

 

y = f(x), the x-axis and the straight line x = a and x = b. The areas above x-axis are taken place plus sign and the areas below x-axis are taken with minus sign i.e,

 

i.e. ∫ba f(x) dx area OLA – area AQM – area MRB + area BSCD

 

Note:         ∫ba f(x) dx, represents algebraic sum of areas means, that if area of function y = f(x)

 

                 is asked between a to b.

 

                => Area bounded = ∫ba |f(x)|dx and not been represented by ∫ba f(x) dx

 

                e.g., If some one asks the area of y = x3 between -1 to 1.

 

                  1130_Area definition.JPG

 

                Then y = x3 could be plotted as;

 

                 ∴ Area = ∫0–1 –x3 dx + ∫10 x3 dx = 1/2

 

                or, using above definition Area = ∫1–1 |x3| dx = 2 ∫10 x3 dx 

 

                                 = 2 [x4 / 4]10 = 1/2

 

                                 But if, we integrate x3 between -1 to 1.

 

                                 =>  ∫10 x3 dx = 0     which does not represent area.

 

                Thus, students are adviced to make difference between area and definite Integral.

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