Applied Mathematics - Integrals & Application of Integrals

Integrals

• Integration as inverse process of differentiation

• Integration of a variety of functions by substitution, by partial fractions and by parts

• Evaluation of simple integrals of the following types and problems based on them

• Definite integrals as a limit of a sum, Fundamental Theorem of Calculus (without proof)

• Basic properties of definite integrals and evaluation of definite integrals

Applications of the Integrals

• Applications in finding the area under simple curves, especially lines, circles/parabolas/ellipses (in standard form only)

• Area between any of the two above said curves (the region should be clearly identifiable)

SUMMARY

Integrals

1. Integration is the inverse process of differentiation. In the differential calculus, we are given a function and we have to find the derivative or differential of this function, but in the integral calculus, we are to find a function whose differential is given. Thus, integration is a process which is the inverse of differentiation. These integrals are called indefinite integrals or general integrals, C is called constant of integration. All these integrals differ by a constant.

2. From the geometric point of view, an indefinite integral is collection of family of curves, each of which is obtained by translating one of the curves parallel to itself upwards or downwards along the y-axis.

3. Some properties of indefinite integrals are as follows:

i. ∫ [f(x) + g(x)]dx = ∫ f(x) dx + ∫ g(x) dx 

ii. For any real number ∫ k f(x) dx = k ∫ f(x) dx

More generally, if f 1 , f 2 , f 3 , ... , f n are functions and k1 , k2 , ... ,kn are real numbers.

4. A change in the variable of integration often reduces an integral to one of the fundamental integrals. The method in which we change the variable to some other variable is called the method of substitution. When the integrand involves some trigonometric functions, we use some well known identities to find the integrals. Using substitution technique, we obtain the following standard integrals.

(i) ∫ tan x dx = log |sec x| + C                          (ii) ∫  cot x dx = log |sin x| + C

(iii) ∫ sec x dx = log |sec x + tan x| + C          (iv) ∫ cosec x dx = log |cosec x - cot x| + C

Applications of the Integrals

1. The area of the region bounded by the curve y = f (x), x-axis and the lines x = a and x = b (b > a)

2. The area of the region bounded by the curve x = φ (y), y-axis and the lines y = c, y = d