NCERT Solutions for Class 12 Maths Chapter 7 Exercise 7.8 Integrals - PDF Download

1. Definite Integral

The definite integral has a unique value. A definite integral is denoted by $\int_a^b f(x) d x$ , where a is called the lower limit of the integral and b is called the upper limit of the integral. The definite integral is introduced either as the limit of a sum or if it has an antiderivative F in the interval [a, b], then its value is the difference between the values of F at the end points, i.e., F(b) – F(a). 

2. Fundamental Theorem of Calculus

• Area function

We have defined $\int_a^b f(x) d x$ as the area of the region bounded by the curve y = f (x), a ≤ x ≤ b, the x-axis and the ordinates x = a and x = b. Let x be a given point in [a, b]. Then $\int_a^x f(x) d x$ represents the Area function A (x).

The fundamental theorems of integral calculus are derived from this concept of area function.

• First fundamental theorem of integral calculus 

Theorem 1: Let f be a continuous function on the closed interval [a, b] and let A(x) be the area function. Then A′(x) = f (x), for all x ∈ [a, b].

• Second fundamental theorem of integral calculus

Below, we provide a significant theorem that allows us to use antiderivatives to evaluate definite integrals.

Theorem 2: Let f be a continuous function defined on the closed interval [a, b] and F be an antiderivative of f. Then $\int_a^b f(x) d x=[\mathbf{F}(x)]_a^b=\mathbf{F}(b)-\mathbf{F}(a)$

Important Points

(i) In words, Theorem 2 tells us that $\int_a^b f(x) d x$ = (value of the anti derivative F of f at the upper limit b – value of the same anti derivative at the lower limit a).

(ii) This theorem is very useful, because it gives us a method of calculating the definite integral more easily, without calculating the limit of a sum.

(iii) The crucial operation in evaluating a definite integral is that of finding a function whose derivative is equal to the integrand. This strengthens the relationship between differentiation and integration.

(iv) In $\int_a^b f(x) d x$ , the function f needs to be well defined and continuous in [a, b].

Tips for Solving Exercise 7.8 Class 12 Chapter 7 Integrals

Class 12 maths chapter 7 exercise 7.8 NCERT solutions offers a variety of questions based on calculating the antiderivative of definite integrals and basic calculus theorems. Our subject experts of eSaral have provided some useful tips for solving ex 7.8 class 12 maths chapter 7.

1. Before solving ex 7.8 class 12 maths chapter 7, you must comprehend the definite integrals as the limit of sums.

2. There are two important theorems based on integral calculus which you must study for better understanding of concepts.

3. To avoid making mistakes in calculations, students should strive to adopt a comprehensive learning strategy. 

Importance of Solving Ex 7.8 Class 12 Maths Chapter 7 Integrals

You will get numerous benefits, by solving questions in ex 7.8 class 12 maths NCERT solutions. eSaral’s experts of mathematics have combined some of the benefits here that can be checked below.

1. This exercise has questions based on significant topics such as definite integral and fundamental theorems of calculus which are explained in simple language by subject experts of eSaral.

2. All the questions in ex 7.8 class 12 maths chapter 7 are solved in stepwise manner to help you understand the concepts used in each question.

3. Practicing and revising the questions will help you to remember the theorems and methods that will help you to solve questions asked in exams.

4. Answers provided in NCERT solutions for the questions of ex 7.8 class 12 maths are accurate as they are solved by experienced teachers of eSaral which can be trusted without any doubt.
   

Frequently Asked Questions

Question 1. What is a definite integral?

Answer 1. The definite integral is introduced either as the limit of a sum or if it has an antiderivative F in the interval [a, b], then its value is the difference between the values of F at the end points. The definite integral has a unique value. A definite integral is denoted by $\int_a^b f(x) d x$ , where a is called the lower limit of the integral and b is called the upper limit of the integral.

Question 2. Can a definite integral be zero?

Answer 2. Yes, the value of a definite integral can be positive, negative and zero.