Definite integrals are used when the limits are defined, to generate a unique value. Indefinite integrals are implemented when the limits of the integrand are not specified. In case, the lower limit and upper limit of the independent variable of a function is specified, its integration is described using definite integrals. A definite integral is denoted as:
F(b) – F(a) = a∫b f(x) dx
where a is the lower limit and b is the upper limit.
f(x)is the integrand.
dx is the integrating agent.
The equation indicates the integral of f(x) with respect to x.
Definite Integral Definition
The definite integral of any function can be expressed either as the limit of a sum or if there exists an anti-derivative F for the interval [a, b], then the definite integral of the function is the difference of the values at points a and b. Let us discuss definite integrals as a limit of a sum.
Definite Integral as the Limit of a Sum and its Formula
Consider a continuous function f in x defined in the closed interval [a, b]. Assuming that f(x) > 0, the following graph depicts f in x.
The integral of f(x) is the area of the region bounded by the curve y = f(x). This area is represented by the region ABCD as shown above. This entire region lying between [a, b] is divided into n equal sub intervals given by [x0, x1], [x1, x2],…… [xr-1, xr],[xn-1, xn]. Considering the width of each sub interval to be h such that h → 0, x0 = a, x1 = a + h, x2 = a + 2h,…..,xr = a + rh,xn = b = a + nh. Here n→∞.
From the above figure,
Area of rectangle PQFR < area of the region PQSRP < area of rectangle PQSE …………(1)
Since. h→ 0, therefore xr– xr-1→ 0. Following sums can be established as;
From the first inequality, considering any arbitrary subinterval [xr-1, xr] where r = 1, 2, 3….n, it can be said that, sn< area of the region ABCD <Sn
Since, n→∞, the rectangular strips are very narrow, it can be assumed that the limiting values of sn and Sn are equal and the common limiting value gives us the area under the curve, i.e.,
From this, it can be said that this area is also the limiting value of an area lying between the rectangles below and above the curve. Therefore,
Here, (b-a)/n → 0, when n → ∞.
This is defined as the definite integral as the limit of a sum.
Properties of Definite Integral
There are some properties of definite integral which could help to evaluate the problems based on it, easily.
- ∫ab f(x) dx = ∫ab f(t) d(t)
- ∫ab f(x) dx = – ∫ba f(x) dx
- ∫aa f(x) dx = 0
- ∫ab f(x) dx = ∫ac f(x) dx + ∫cb f(x) dx
- ∫ab f(x) dx = ∫ab f(a + b – x) dx
- ∫0a f(x) dx = f(a – x) dx
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