Consider a curve which is above the x – axis. It is a continuous function on an interval [a, b] where all the values are positive. The area between the curve and the x-axis defines the definite integral
of any continuous function. In the above formulae, a and b are the limits, d/dx (F(x)) = f (x).
The definite integral is different from the indefinite integral as follows:
Indefinite integral lays the base for definite integral.
Indefinite integral defines the calculation of indefinite area whereas definite integral is finding area with specified limits.
The following are the properties of definite integrals
1. ∫ab f (x) dx = ∫ab f (t) dt
2. ∫ab f (x) dx = – ∫ba f (x) dx … [Also, ∫aa f(x) dx = 0]
3. ∫ab f (x) dx = ∫ac f (x) dx + ∫cb f (x) dx
4. ∫ab f (x) dx = ∫ab f (a + b – x) dx
5. ∫0a f (x) dx = ∫0a f (a – x) dx …
6. ∫02a f (x) dx = ∫0a f (x) dx + ∫0a f (2a – x) dx
7. Two parts
∫02a f (x) dx = 2 ∫0a f (x) dx … if f (2a – x) = f (x).
∫02a f (x) dx = 0 … if f (2a – x) = – f(x)
8. Two parts
∫-aa f(x) dx = 2 ∫0a f(x) dx … if f(- x) = f(x) or it is an even function
∫-aa f(x) dx = 0 … if f(- x) = – f(x) or it is an odd function
9. Walli’s formula
∫0π/2sinnxcosmxdx=[(n−1)(n−3)(n−5)….1or2][(m−1)(m−3)(m−5)….1or2].K/(m+n)(m+n−2)….1or2) , provided if both m and n are even, then K = π/2
10. Leibnitz’s rule
If g is continuous on [a, b] and f1 (x) and f2 (x) are the two differentiable functions whose values lie in [a, b], then d / dx ∫ f2(x) f1(x) g(t) dt = g (f2(x)) f2′(x) – g (f1(x)) f1′(x)
How to Measure Definite Integral
The area under the graph is the definite integral. By definition, definite integral is the sum of the product of the lengths of intervals and the height of the function that is being integrated with that interval, which includes the formula of the area of the rectangle. The figure given below illustrates it.