NCERT Solutions for Class 12 Maths Chapter 8 – Application of Integrals
NCERT Solutions for Class 12 Maths Chapter 8 – Application of Integrals, prepared by Myclass24, focus on helping students use integration as a tool to calculate areas enclosed by curves, lines, and axes. After learning the mechanics of integration in Chapter 7, this chapter brings a geometric dimension to the topic. Our solutions are written to make the process of setting up the integral — which is often the hardest part — completely transparent and easy to follow.
Summary of Chapter 8 Applications of Integrals
Integration of irrational functions, Reduction formula, introduction- definite integral as the limit of a sum, Evaluation of integrals by substitution
Definite integral as the limit of a sum, Evaluation of integrals by substitution. Properties of definite integrals, Estimation of integrals, Gamma Function, Reduction formula. Introduction, Area under the simple curve, Curve sketching. Basic problems. The area under the curves.
Find the PDF of NCERT Solutions for Class 12 Maths Chapter 8
The PDF of NCERT Solutions for Class 12 Maths Chapter 8 – Application of Integrals is available on Myclass24. Students can access all exercise-wise solutions in a clean, printable format. The PDF is structured to match the textbook order, making it easy to follow along while studying. Download and keep it handy during your revision sessions for quick reference before tests and board
Detail | Information |
Chapter | Chapter 8 – Application of Integrals |
Class | Class 12 Maths |
Board | CBSE (NCERT) |
Total Exercises | 2 Exercises + Miscellaneous |
Topics Covered | Area Under Curves, Area Between Two Curves, Area Using Parabolas, Ellipses and Lines |
Difficulty Level | Moderate |
Content By | Myclass24 |
About Chapter 8 – Application of Integrals
Chapter 8 is relatively compact compared to Chapter 7, but the problems here require a strong ability to visualise curves on a coordinate plane. The core idea is simple: the area between a curve y = f(x) and the x-axis from x = a to x = b is given by the definite integral of f(x) from a to b. But making that formula work on actual problems involves several layers of thinking.
Exercise 8.1-students begin with standard curves such as circles, parabolas, and ellipses. A common question asks for the area enclosed by the circle x² + y² = r², and students need to identify the correct limits, handle the square root correctly, and integrate — all while keeping track of symmetry to simplify the work.
The concept of symmetry is used heavily in this chapter. For a circle or an ellipse, integrating over one quadrant and multiplying by four is a standard approach that our solutions at Myclass24 demonstrate clearly in every relevant problem.
Exercise 8.2 introduces the idea of the area between two curves. Here, the integral is set up as the integral of the difference (upper curve minus lower curve), and the limits are determined by the points of intersection of the two curves. Finding those intersection points — by solving the two equations simultaneously — is a step that students sometimes rush through, leading to errors in the limits.
The Miscellaneous Exercise combines area problems involving lines and curves together. For instance, a region might be bounded partly by a parabola and partly by a straight line, requiring students to split the integral at the point of intersection. Our solutions break these into neat sub-problems and present them in a format that works well for board exam answers.
The chapter as a whole reinforces the link between algebra, geometry, and calculus — making it an important part of the Class 12 curriculum at Myclass24.
Why Study Chapter 8 from Myclass24?
At Myclass24, every solution for Chapter 8 is crafted keeping the CBSE exam pattern in mind. The explanations are written in plain language, the steps are numbered clearly, and important formulas are highlighted where needed. Students preparing for board exams, or anyone who wants to genuinely understand Application of Integrals, will find Myclass24 solutions both reliable and thorough.
NCERT Solutions for Class 12 Maths Chapter 8 – Application of Integrals FAQs
Chapter 8 introduces students to the practical use of integration in geometry and mathematical analysis. The primary focus is on finding the area of regions bounded by curves, straight lines, and coordinate axes. Unlike the previous chapter, which concentrates on integration techniques, this chapter explains how those techniques are applied to solve real-world mathematical problems. NCERT exercises contain a variety of questions involving graphical interpretations and area calculations. Understanding these applications helps students connect calculus with geometry and develop a deeper understanding of mathematical relationships. This chapter is also important because area-based questions frequently appear in board examinations.
Finding the area under a curve is one of the most important concepts in this chapter. Students first identify the function, determine the interval over which the area is required, and then evaluate the definite integral within those limits. The resulting value represents the area enclosed by the curve and the coordinate axis. NCERT exercises include several examples involving polynomial, trigonometric, and other functions. Careful attention must be paid to the limits of integration and the graphical representation of the region. Regular practice helps students understand the geometric meaning of integration and improves their accuracy in solving area-related problems.
The area between two curves is calculated by integrating the difference between the upper curve and the lower curve over a specified interval. Students must first identify which curve lies above the other and determine the points where the curves intersect. NCERT questions often test the ability to visualise graphs and correctly establish integration limits. A diagram is extremely helpful in understanding the enclosed region. Once the limits and functions are identified correctly, the area can be calculated systematically. Mastering this concept helps students solve complex geometry-based questions and strengthens their understanding of the relationship between graphs and integrals.
Graphical interpretation plays a crucial role in this chapter because area calculations depend on the shape and position of curves. Students who sketch graphs before solving problems often find it easier to identify boundaries, intersections, and integration limits. NCERT exercises frequently involve regions enclosed by multiple curves, making visualisation an essential skill. A graph helps prevent mistakes such as selecting incorrect limits or subtracting functions in the wrong order. Developing graphical understanding not only improves accuracy but also makes mathematical concepts more intuitive. This approach allows students to solve application-based questions more confidently and efficiently.
A common mistake is using incorrect limits of integration, which can completely change the final answer. Students also sometimes fail to identify which curve lies above the other when calculating the area between curves. Errors in graph interpretation and algebraic simplification are also frequent. Another issue is neglecting to sketch the region before beginning calculations. NCERT exercises encourage students to adopt a systematic approach by first drawing diagrams and then applying integration methods carefully. Regular practice helps eliminate these errors and improves conceptual understanding. Accuracy in visualisation and calculation is the key to success in this chapter.