Class 12 Maths - CBSE
Application of Integrals
The chapter 'Application of Integrals' in Class 12 CBSE Mathematics teaches you how to use definite integrals to calculate the exact areas enclosed by plane curves, lines, circles, parabolas, and ellipses. Building directly upon your integration skills from the previous chapter, this topic bridges abstract calculus with geometric visualization. It is a high-scoring chapter in the CBSE board exams, frequently featuring standard 5-mark long-answer questions that require careful graph sketching and proper boundary identification. Mastering this chapter will strengthen your graphical intuition and problem-solving abilities for both board exams and competitive tests like JEE.
Start Learning FreeKey Concepts
Area under a simple curve
The area bounded by the curve y = f(x), the x-axis, and the ordinates x = a and x = b is given by the definite integral from a to b of y dx.
Area between two curves
The area enclosed between two intersecting curves y = f(x) and y = g(x) is found by integrating the difference of the upper and lower functions between their points of intersection.
Symmetry in curves
Identifying if a curve is symmetrical about the x-axis, y-axis, or origin helps you calculate only a portion of the area and multiply it accordingly, simplifying calculations.
Rough Sketching
Drawing an accurate rough sketch of standard curves like circles, parabolas, ellipses, and lines is crucial for determining the correct limits of integration.
Important Formulas
Board Exam Info
In the CBSE Class 12 Mathematics exam, this chapter typically carries around 6 to 8 marks. Questions usually include one compulsory long-answer question worth 5 marks involving finding the area enclosed between a line and a parabola or circle, along with a 1-mark or 2-mark objective or short-answer question.
Frequently Asked Questions
Is drawing a rough sketch compulsory in the exam?
Yes, drawing a clear rough sketch showing the curves, intersection points, and shaded region is mandatory. Examiners award marks specifically for the correct graph and shading.
How do I decide whether to integrate with respect to x or y?
Choose dx if you are using vertical strip elements (y in terms of x) or dy if you are using horizontal strip elements (x in terms of y), depending on which makes finding limits and integration easier.
Do I need to include the modulus sign or consider negative areas?
Definite integrals can yield negative values if the area lies below the x-axis. Since area is always positive, you must take the absolute value or split the integral at the x-axis intercepts.
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