Fundamental Theorem of Calculus

The fundamental theorem of calculus is a theorem that links the concept of integrating a function with that differentiating a function.

When the upper and the lower limit of an independent variable of the function or integrand is, its integration is described by definite integrals. It is expressed as:

Here R.H.S. of the equation indicates integral of f(x) with respect to x.

f(x) is the integrand.

dx is the integrating agent.

‘a’ indicates upper limit of the integral and ‘b’ indicates lower limit of the integral.

Function of a definite integral has a unique value. Definite integral of a function can be described as a limit of a sum. If there is an anti-derivative F of the function in the interval [a, b], then the definite integral of the function is the difference between the values of F, i.e., F(a) – F(b).

Fundamental Theorem of Calculus

Lets consider a function f in x that is defined in the interval [a, b]. The integral of f(x) between the points a and b i.e. $\int_{ a }^{ b } f(x)d(x)$,  is the area of that is bounded by the curve y = f(x) and the lines x = a, x =b and x – axis $\int_{a}^{x} f(x)dx$.  depicts the area of the region shaded in brown where x is a point lying in the interval [a, b]. Assuming that the values taken by this function are non- negative, the following graph depicts f in x.

A(x) is known as the area function which is given as;

Depending upon this, the fundamental theorem of Calculus can be defined as two theorems as stated below;

First Fundamental Theorem of Calculus

Theorem 1: If f is a continuous function defined on the closed interval [a, b] and A(x) is the area function, then

A’(x) = f(x) x [a, b]

Second Fundamental Theorem of Calculus

Theorem 2: If f is a continuous function defined on the closed interval [a, b] and F denotes the anti-derivative of f, then

This theorem gives the area bounded by the curve f(x) lying between the closed interval [a, b] as the difference of the value of anti-derivative F of f at the upper limit b and value of anti-derivative F of f at the lower limit a.

Steps for calculating $\int_{a}^{b} f(x)dx$

• Determine the indefinite integral of f(x) as F(x). It must be noticed that arbitrary constant is not considered while calculating definite integrals since it cancels out itself, i.e.,
• Calculate F(b) – F(a) which gives us the value of the definite integral of f in x lying between the closed interval [a, b].

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Practise This Question

A cricket ball of radius r is exactly fitted in a cylindrical tin as shown in the figure below. The ratio of surface areas of cricket ball to the surface area tin is