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# RS Aggarwal Class 12 Chapter 16 Solutions (Definite Integrals)

RS Aggarwal Solutions for Class 12 Chapter 16- Definite Integrals are prepared to make you proficient enough in this topic so that you can solve all the questions effortlessly. You will be introduced to definite integrals in this chapter. Some other topics that will be covered in the exercise of this chapter of RS Aggarwal Solutions for Class 12 are Definite Integral as the Limit of a Sum, Irrational or Rational expressions of Definite Integral and properties of the definite integrals. Solving these exercise questions will assist you in preparing for your higher studies besides helping you get good marks in your school exams.

There are 4 exercises in Chapter 16 comprising a total of 94 questions to provide you with ample resource material for rigorous practice. The exercises have a diverse set of questions ensuring the robust coverage of all the topics in the CBSE board syllabus. You will get a rough idea about the pattern of questions asked in the exams after solving these questions and therefore, you can be prepared in advance for your exams.

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## Topics Covered in RS Aggarwal Solutions for Class 12 Chapter 16- Definite Integrals

Introduction:

A definite integral of any function can be shown as a limit of the sum or if there is an anti-derivative F in the given closed interval [a, b], then the integral of the function is the difference of the values at points a and b. Here, we will be discussing definite integrals as a limit of a sum.

Definite Integral as Limit of a Sum:

Let f be a function in the closed interval [a, b] which is continuous in x. The integral of f(x) is the area of the region covered under the curve y = f(x). This entire region lying between [a, b] is divided into n equal sub-intervals given by [x0, x1], [x1, x2],… [xr-1, xr], [xn-1, xn]. Taking the width of every sub-interval as h such that   h → 0, x0 = a, x1 = a + h, x2 = a + 2h,…..,xr = a + rh, xn = b = a + nh. Here n→∞.

Since. h→ 0, therefore xr –  xr-1 → 0. Following sums can be established as;

Since, n→∞, it can be assumed that the limiting values of sn and Sn are equal and the common limiting value gives us the area under the curve, i.e.,

From the above, we can say that this area is also the limiting value of the area lying between the rectangles above and below the curve. Hence,

Here, (b-a)/n → 0, when n → ∞.

The above formula is known as the definite integral as the limit of a sum.

Definite Integrals Rational or Irrational Expression:

Highlights:

Properties of Definite Integral: There are some properties of definite integral which could help to evaluate the problems based on it, easily.

• ∫ab f(x) dx = ∫ab f(t) d(t)
• ∫ab f(x) dx = – ∫ba f(x) dx
• ∫aa f(x) dx = 0
• ∫ab f(x) dx = ∫ac f(x) dx + ∫cb f(x) dx
• ∫ab f(x) dx = ∫ab f(a + b – x) dx
• ∫0a f(x) dx =  f(a – x) dx

### Important Topics Mentioned in RS Aggarwal Solutions for Class 12 Chapter 16 – Definite Integrals

1. There is a broad range of questions in the exercises of this chapter covering all the topics of this chapter.
2. The exercise questions in RS Aggarwal Class 12 Solutions for Chapter 16 will assist you in preparing for high-level pan India exams such as JEE, NEET, BITSAT etc.
3. Solving these questions will consistently help you increase your skills to manage your time in the high-pressure environment of the exam hall.
4. The exercise questions are arranged in an orderly fashion of increasing complexity level so as to make the concepts interesting and compatible with your level of understanding.
5. The exercises are well synchronised with the pattern of question papers of all the important exams of your level.

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