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 x-axis 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 x-axis.

In general, 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.

∴

where A1, A2, A3, A4, A5 are the areas of the shaded region.

Properties of definite integrals

1. Change of dummy variable:

2. Interchanging limits:

3. Split of limits:

where 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.

Important result

=

5. If f(x) is discontinuous at x = a, then
6. =
7. If f(t) is an odd function then φ(x) = is an even function
8. dx = ndx; where “T” is the period of the function and n I, i.e., f(x + T) = f(x)
9. f(x) dx = (nm) f(x) dx, where “T” is the period of the function and m, n I
10. f(x) dx = where “T” is the period of the function and n I