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Geometrical Interpretation of the Definite Integral

First we construct the graph of the integrand y = f(x) then in the case of f(x) ≥ 0 x [a, b], the integral 98499.png is numerically equal to the area bounded by the curve y = f(x), the x-axis and the ordinates of x = a and x = b.
 
99415.png
 
98493.png is numerically equal to the area of curvilinear trapezoid bounded by the given curve, the straight lines x = a and x = b, and the x-axis.
 
In general, 98487.png represents an algebraic sum of areas of the region bounded by the curve y = f(x), the x-axis and the ordinates x = a and x = b.
 
The areas above the x-axis enter into this sum with a positive sign, while those below the x-axis enter it with a negative sign.
 
98481.png
 
∴ 98475.png
 
where A1, A2, A3, A4, A5 are the areas of the shaded region.

Properties of definite integrals

  1. Change of dummy variable:
     
    98469.png
  2. Interchanging limits:
     
    98463.png
  3. Split of limits:
     
    98456.pngwhere c may lie inside or outside the interval [a, b]. This property to be useful when function is in piecewise definition for x (a, b) or when f(x) is discontinuous or non-differentiable at x = c.
  4. 98449.png
  5. 98443.png
     
    98436.png

     

    Important result

     

    98429.png = 98423.png98417.png

  6. If f(x) is discontinuous at x = a, then 98411.png
  7. 98405.png
  8. 98399.png= 98393.png
  9. If f(t) is an odd function then φ(x) = 98387.pngis an even function
  10. 98381.pngdx = n98374.pngdx; where “T” is the period of the function and n I, i.e., f(x + T) = f(x)
  11. 98368.pngf(x) dx = (nm) 98362.pngf(x) dx, where “T” is the period of the function and m, n I
  12. 98356.pngf(x) dx = 98350.png where “T” is the period of the function and n I




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