JEE Exam  >  JEE Notes  >  Mathematics (Maths) for JEE Main & Advanced  >  Definite Integration and Its Properties

Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced PDF Download

Definite Integral Definition

The definite integral of a real-valued function f(x) with respect to a real variable x on an interval [a, b] is expressed as

Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced

Here,

∫ = Integration symbol

a = Lower limit

b = Upper limit

f(x) = Integrand

dx = Integrating agent

Thus, ∫ab f(x) dx is read as the definite integral of f(x) with respect to dx from a to b.

Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced

The Area Problem

Use rectangles to estimate the area under the parabola y = x2 from, 0 to 1 Wr first notice that the area of S must be somewhere between 0 and 1 because S is contained in a square with side length 1, but we can certainly do better than that. Suppose we divide S into four strips S1, S2, S3, and S4 by drawing the vertical lines x = 1/4, x = 1/2, x = 3/4,as in Figure (a) 

Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced

Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced

We can approximate each strip by a rectangle whose base is the same as the strip and whose height is the same as the right edge of the strip [see Figure (b)]. In other words, the heights of this rectangle are the values of the function f(x) = x2 at the right endpoints of the subintervals  Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced  Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced

Each rectangle has a width of 1/4th and the heights are  Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced  and 12. If we let R4 be the sum of the areas of these approximating rectangles, we get

Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced

From Figure (b) we see that the area A of S is less than R4, so A < 0.46875 Instead of using the rectangles in Figure (b) we could use the smaller rectangles in Figure (c) whose heights are the values of f at the left endpoints of the subintervals. (The leftmost rectangle has collapsed because its heights is 0). The sum of the areas of these approximating rectangles is

Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced

We see that the area of S is larger than L4, so we have lower and upper estimates for A , 0.21875 < A < 0.46875 We can repeat this procedure with a larger number of strips. Figure (d), (e) shows what happens when we divide the region S into eight strips of equal width. By computing the sum of the areas of the smaller rectangles (L8) and the sum of the areas of the larger rectangles (R8), we obtain better lower and upper estimates for A :  0.2734375 < A < 0.3984375

Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced

Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced

So one possible answer the question is to say that the true area of S lies somewhere between 0.2734375 and 0.3984375. We coluld obtain better estimates by increasing the number of strips. 


Properties Of Definite Integral

Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced


1. Change of Variable: The definite integral Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced  is a number, it does not depend on x. In fact, we could use any letter in place of x without changing the value of the integral ;
Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced =  Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced

2. Change of Limit: When we defined the definite integral Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced f(x)dx, we implicity assumed that a < b. But the definition as a limit of sum makes sense even if a > b. Notice that if we reverse a and b, then Δx changes from (b – a)/n to (a – b)/n.

Therefore  Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced


3. Additivity With Respect To The Interval Of Integration :

Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced

Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced

This is not easy to prove in general, but for the casewhere f(x) ≥ 0 and a < c < b Property 7 can be seen from the geometric interpretation in Figure : The area under y = f(x) from a to c plus the area from c to b is equal to the total  area from a to b.

Solved Examples

Ex.1 If it is known that Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced


Sol. We have

Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced

= 17 -12 =5
Ex.2 Find  Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced


Sol. The function f defined by f(x) = 3 – 2x + x2 is continuous and has antiderivative g defined by
Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced  Therefore, by the fundamental theorem of the calculus,
Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced

Ex.3 Evaluate  Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced

Sol. We can rewirte the integrand as follows  Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced

From this, you can rewrite the integral in two parts.
Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced

Ex.4 Evaluate Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanceddx, (where [ * ] denotes greatest integer function)

Sol.Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced

Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced

Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced

Ex.5 Find the range of the function f(x) = Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced


Sol. Given that

Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced 

Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced

Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced


Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced

Proof : Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced

In the first integral on the far right side we make the substitution u = –x. Then du = –dx and when x = –a, u = a. 

Therefore   Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced   Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced

(a) If f is even, then  Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced

(b) If f is odd, then Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced

Theorem is illustrated by Figure(a, b) For the case where f is positive and even, part(a) says that the area under y= f(x) from –a to a is twice the area from 0 to a because of symmetry. Thus, part (b) says the integral is 0 because the areas cancel.

Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced

Since f(x) = x6 + 1 satisfies f(–x) = f(x), it is even and so

Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced  Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced

Since f(x) = (tan x)/(1 + x2 + x4) satisfies f(–x) = –f(x), it is odd and so  Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced


Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced


Ex.6 Evaluate  Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced


Sol. The integrand, f(x) = x3 – 3x, is an an odd function; i.e., the equation f(–x) is satisifed for every x. Its graph, drwan in Figure, is therefore symmetric under reflection first about the x-axis and then about the y-axis. If follows that the region above the x-axis has the same areas as the region below it.

Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced

We conclude that Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced = 0.


Ex.7 Evaluate  Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced where [x] denotes the greatest integer function less than or equal to x.

Sol.

Let Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced  Suppose f(x) = (–1)[x]

∴ f(–x) = (–1)[–x] = (–1)–1–[x], x ∉ l = - (-1)-[x]

Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced = – f(x), f(x) is odd function.

Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced


Ex.8 Show that  Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced

Sol.

Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced  (By prop.) ........(1)

Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced

adding (1) and (2) we get 2l  Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced

Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced (By prop.)  Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced

Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced ...(3)

Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced

Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced ...(4)

adding (3) and (4) we get Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced


Ex.9 Evalute Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced

Sol.

Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced ....(1)

Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced

Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced

Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced

Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced

Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced

Adding (1) and (2) we get 2l  Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced

Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced

Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced

Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced

Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced

Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced
 
Ex.10 Evaluate Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced (where { * } denotes the fractional part function)

Sol.

Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced

Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced

Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced = 1(0 + 1) + 15(1 – 0) = 16 (∴ {x} is a periodic function)


Ex.11 Let the function f be defined by f(x) = Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced
 Evaluate  

Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced

Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced


Sol.

Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced

Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced


Ex.12 Evaluate Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced  where [x] and {x} denotes the integral part, and fractional part function of x and n ε N.

Sol.

Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced

Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced

Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced

 

Ex.13 Show that  Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced

Sol.

Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced

Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced

Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced

Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced

Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced

= q{(1 – 0) – (0 – 1)} + sin p – sin 0 = 2q + sin p

 

Ex.14 Evaluate  Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced   (where [ * ] is the greatest integer function)


Sol.

Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced

Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced

Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced

Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced

Since sin x + cos x is periodic function with period 2π, so
Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced

Alternative Method : (Graphical Method)

It is clear from the figure.

Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced

Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced

Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced

Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced

The document Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced is a part of the JEE Course Mathematics (Maths) for JEE Main & Advanced.
All you need of JEE at this link: JEE
209 videos|443 docs|143 tests

Up next

FAQs on Definite Integration and Its Properties - Mathematics (Maths) for JEE Main & Advanced

1. What is the definition of a definite integral?
Ans. A definite integral is a mathematical concept used in calculus to determine the area under a curve between two points on a graph. It represents the accumulation of values of a function over a specific interval.
2. What are the properties of a definite integral?
Ans. The properties of a definite integral include linearity, additivity, and the property of reversing limits. Linearity means that the integral of a sum of functions is equal to the sum of their integrals. Additivity means that the integral of a function over a union of intervals is equal to the sum of the integrals over each individual interval. The property of reversing limits states that changing the order of the limits in a definite integral does not affect its value.
3. How can definite integration be used to solve real-world problems?
Ans. Definite integration can be used to solve a variety of real-world problems, such as finding the area between curves, calculating displacement and velocity of objects, determining the total accumulated value of a quantity over time, and evaluating probabilities in statistics. It provides a powerful tool for analyzing and quantifying various physical and mathematical phenomena.
4. What is the difference between a definite integral and an indefinite integral?
Ans. The main difference between a definite integral and an indefinite integral is that a definite integral has specific limits of integration, representing the interval over which the function is integrated. It gives a numerical value as the result. On the other hand, an indefinite integral does not have limits and represents the antiderivative of a function. It gives a general expression involving a constant as the result.
5. Can the properties of definite integration be applied to improper integrals?
Ans. Yes, the properties of definite integration can be applied to improper integrals as well. Improper integrals involve functions that may be infinite or undefined at certain points within the interval of integration. The properties of linearity, additivity, and reversing limits still hold true for improper integrals, allowing for the application of these properties to simplify calculations and solve problems involving such integrals.
209 videos|443 docs|143 tests
Download as PDF

Up next

Explore Courses for JEE exam
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
10M+ students study on EduRev
Related Searches

ppt

,

practice quizzes

,

Important questions

,

Objective type Questions

,

Summary

,

study material

,

Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced

,

video lectures

,

Semester Notes

,

past year papers

,

Free

,

Previous Year Questions with Solutions

,

Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced

,

MCQs

,

Viva Questions

,

mock tests for examination

,

pdf

,

Exam

,

Extra Questions

,

Definite Integration and Its Properties | Mathematics (Maths) for JEE Main & Advanced

,

Sample Paper

,

shortcuts and tricks

;