NCERT Class 12 Mathematics Mathematics Part-II: Chapter 2 — MATHEMATICS
This chapter, Application of Integrals, focuses on using integral calculus to find the areas enclosed by curves. Building upon the concept of definite integrals as a limit of sums from the previous chapter, it introduces methods to calculate areas under simple curves, between lines and arcs of circles, parabolas, and ellipses. The chapter explains how to find the area bounded by a curve y = f(x), the x-axis, and vertical lines x = a and x = b, by summing up elementary vertical strips of area ydx. It also covers finding the area bounded by a curve x = g(y), the y-axis, and horizontal lines y = c and y = d using horizontal strips. The text clarifies how to handle areas where the curve lies below the x-axis by taking the absolute value of the negative integral, and how to combine areas above and below the x-axis for the total bounded area. This chapter is crucial for developing problem-solving skills in geometry and calculus for CBSE Class 12 students.
Quick info
| Board | CBSE / NCERT |
|---|---|
| Class | Class 12 |
| Subject | Mathematics |
| Book | Mathematics Part-II |
| Chapter | Chapter 2 — MATHEMATICS |
| Language | English |
| PDF type | NCERT Textbook |
| Session | CBSE 2026 |
| Reading time | 4 minutes |
| Word count | 609 |
Learning outcomes
- Understand the concept of finding areas using definite integrals.
- Calculate the area under simple curves bounded by the x-axis and given ordinates.
- Determine the area bounded by a curve and the y-axis.
- Apply integral calculus to find areas of regions bounded by standard curves like circles, parabolas, and ellipses.
- Handle cases where the curve lies below the x-axis.
Vocabulary
| Word | Meaning |
|---|---|
| Integral Calculus | A branch of calculus concerned with the summation of quantities; it is the reverse of differential calculus. |
| Definite Integral | An integral that has upper and lower limits, representing the net area under a curve between those limits. |
| Ordinates | Vertical lines, typically represented by x = constant in a Cartesian coordinate system. |
| Elementary Area | A small, fundamental unit of area used in integration, often represented as dA. |
| Parabola | A symmetrical open curve formed by the intersection of a cone with a plane parallel to its side. |
| Ellipse | A closed curve, the set of points such that the sum of the distances from two fixed points (foci) is constant. |
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Practice questions
- What is the formula for the area bounded by the curve x = g(y), the y-axis, and the lines y = c and y = d? Answer: A = ∫[from c to d] g(y) dy
- If a curve lies below the x-axis between x = a and x = b, how is the bounded area calculated? Answer: The area is the absolute value of the definite integral: |∫[from a to b] f(x) dx|.
- What fundamental theorem is used to evaluate definite integrals? Answer: The Fundamental Theorem of Calculus.
Practice MCQs
Q1. The area bounded by the curve y = f(x), the x-axis, and the ordinates x = a and x = b is given by:
Explanation: The area under the curve y = f(x) from x = a to x = b is calculated by the definite integral of f(x) with respect to x from a to b.
Q2. If f(x) < 0 for x in [a, b], the definite integral ∫[from a to b] f(x) dx represents:
Explanation: When the curve is below the x-axis, the definite integral yields a negative value, representing the signed area.
Q3. To find the area bounded by the curve x = g(y), the y-axis, and lines y = c, y = d, we integrate with respect to:
Explanation: When the curve is defined as x in terms of y, the area is found by integrating with respect to y between the given y-limits.
Q4. The elementary area of a thin vertical strip is given by:
Explanation: A vertical strip has height y and infinitesimal width dx, so its area dA is ydx.
Q5. What is the primary application of integrals discussed in this chapter?
Explanation: This chapter specifically focuses on using integrals to compute areas bounded by various types of curves.
Frequently asked questions
What is the main purpose of the chapter 'Application of Integrals' in Class 12 Mathematics?
The chapter aims to teach students how to use integral calculus to calculate the areas of regions bounded by various curves, lines, and axes.
How is the area under a curve y = f(x) from x = a to x = b calculated?
It is calculated by finding the definite integral of f(x) with respect to x from a to b, i.e., ∫[from a to b] f(x) dx.
What happens if the curve is below the x-axis?
If the curve is below the x-axis, the definite integral will be negative. The actual area is the absolute value of this negative integral.
Can integrals be used to find areas bounded by curves defined as x in terms of y?
Yes, the area bounded by x = g(y), the y-axis, and horizontal lines y = c and y = d is found by integrating g(y) with respect to y from c to d.
What are 'ordinates' in the context of finding areas?
Ordinates are vertical lines, typically represented by x = a and x = b, which define the boundaries of the region along the x-axis.
What is an 'elementary strip' in this chapter?
An elementary strip is a very thin vertical or horizontal strip used to approximate the area under a curve, with its area being ydx or xdy.
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NCERT Class 12 Mathematics — Mathematics Part-II — Chapter 2 — MATHEMATICS. Verified by NCERT Help Editorial Team. Reviewed on 29 Jul 2026. Last updated 10 Aug 2026.