Definite Integral
The definite integral is introduced either as the limit of a sum or if it has an anti derivative F in the close interval [a, b], and its value is F (b) – F (a).
Definite integral as the limit of a sum:
Let f(x) be a continuous function defined on a closed interval [a, b] and f(x) is greater than zero for all x. Then the integral of f(x) over the closed interval [a, b] is equal to the limit h tends to zero h multiplied by the sum of values of functions at points a, a+h and so on till point a+ (n – 1) h.
Here, h = (b – a)/n
The value of the definite integral of a function over any particular interval depends on the function and the interval but not on the variable of integration that we choose to represent the independent variable.
Hence, the variable of the integration is called dummy variable.
Let x be a given point in [a, b]. Then, the area of the shaded region depends on x.
The area function A (x) is the integral of f(x) with respect to x over the closed interval [a, x].
First Fundamental Theorem of Calculus:
Let f be a continuous function on the closed interval [a, b] and A(x) be the area function. Then the derivative of area function is equal to the function f(x) itself for all x in the closed interval [a, b].
Second Fundamental Theorem of Calculus:
Let f be a continuous function on the closed interval [a, b] and F be an anti derivative of f. Then the integral of f(x) over close interval [a, b] is equal to
F (b) – F (a).
Steps for Calculating Definite Integral:
Step 1: Find the indefinite integral of f(x).
Step 2: Calculate the values of indefinite integral for x = a and x = b.
Step 3: Subtract the value of indefinite integral at a from the value of indefinite integral at b.
Definite integrals can also be evaluated by the substitution for integrand, differential and limits.
There are various properties of definite integrals that are very useful to evaluate definite integrals.
Keywords: Definite Integral using properties, Definite integral as the limit of a sum, Definite integral by substitution
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