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 is numerically equal to the area bounded by the curve y = f(x), the xaxis and the ordinates of x = a and x = b.
is numerically equal to the area of curvilinear trapezoid bounded by the given curve, the straight lines x = a and x = b, and the xaxis.
In general, represents an algebraic sum of areas of the region bounded by the curve y = f(x), the xaxis and the ordinates x = a and x = b.
The areas above the xaxis enter into this sum with a positive sign, while those below the xaxis enter it with a negative sign.
∴
where A_{1}, A_{2}, A_{3}, A_{4}, A_{5} are the areas of the shaded region.
Properties of definite integrals
 Change of dummy variable:
 Interchanging limits:
 Split of limits:

Important result
= =

If f(x) is discontinuous at x = a, then
 =
 If f(t) is an odd function then φ(x) = is an even function
 dx = ndx; where “T” is the period of the function and n ∈ I, i.e., f(x + T) = f(x)
 f(x) dx = (n – m) f(x) dx, where “T” is the period of the function and m, n ∈ I
 f(x) dx = where “T” is the period of the function and n ∈ I