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Definite Integration

 

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

1130_Area definition.JPGe.g., If some one asks the area of y = x3 between -1 to 1.

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.

To read more, Buy study materials of Definite integral comprising study notes, revision notes, video lectures, previous year solved questions etc. Also browse for more study materials on Mathematics here.


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