How to Solve Area Under Curves in Class 12 Maths Exercise 8.1
FAQs on Class 12 Maths Chapter 8 Exercise 8.1 NCERT Solutions (Area Under Curves)
1. What is the correct approach to solving questions from NCERT Class 12 Maths Chapter 8 using these solutions?
To effectively use the NCERT Solutions for Chapter 8, focus on the step-by-step methodology. First, attempt the problem yourself. Then, compare your method with the provided solution, paying close attention to how limits are determined, the correct integral is set up, and the final calculations are performed. This process helps you master the CBSE pattern and avoid common errors.
2. How can I correctly find the area between two curves as per the NCERT solutions for Chapter 8?
The standard method for finding the area between two curves involves these key steps:
- First, draw a rough sketch of both curves to visualise the bounded region.
- Next, find the points of intersection by solving the equations of the curves simultaneously. These points will be your limits of integration (a, b).
- Set up the definite integral as ∫ₐᵇ [f(x) - g(x)] dx, where f(x) is the upper curve and g(x) is the lower curve.
- Finally, integrate and evaluate the expression to find the required area.
3. Are the NCERT Solutions for Class 12 Maths Chapter 8 aligned with the latest CBSE 2025-26 syllabus?
Yes, all solutions provided for 'Application of Integrals' are fully updated to match the latest CBSE 2025-26 syllabus. The methods, formulas, and problem types covered are precisely what is prescribed by the NCERT curriculum, ensuring complete relevance for your board exams.
4. What are the most common mistakes to avoid when solving problems on the area under curves?
Students often make a few common errors in Chapter 8. Be careful to avoid:
- Incorrectly identifying the limits of integration. Always solve for intersection points first.
- Confusing the upper and lower functions, which can lead to a negative area if subtracted incorrectly.
- Making simple algebraic or integration errors during calculation.
- Forgetting to draw a sketch, which makes it difficult to visualise the bounded region and set up the integral correctly.
5. Why is drawing a rough sketch considered the most critical first step for finding the area under a curve?
Drawing a sketch is crucial because it provides a visual representation of the problem. It helps you to correctly:
- Identify the bounded region whose area needs to be calculated.
- Determine which function is the 'upper curve' and which is the 'lower curve'.
- Visually confirm the limits of integration (the x or y values where the region begins and ends).
- Decide whether to integrate with respect to x (vertical strips) or y (horizontal strips), which can simplify the problem.
6. How do the NCERT Solutions for the Miscellaneous Exercise of Chapter 8 differ from the regular exercises?
The NCERT Solutions for the Miscellaneous Exercise in Chapter 8 tackle more complex and application-based problems. Unlike the direct questions in Exercises 8.1 and 8.2, these often require a combination of concepts, such as finding the area of a triangle using integration or solving problems with more intricate curves. They are designed to test a deeper, more holistic understanding of the chapter.
7. When solving for a bounded area, how do I decide whether to integrate with respect to x (dx) or y (dy)?
The choice between integrating with respect to x (using vertical strips) or y (using horizontal strips) depends on the orientation of the curves.
- Use dx (vertical strips) when it is easy to express the functions in the form y = f(x) and one function is consistently above the other. This is the most common approach.
- Use dy (horizontal strips) when it is easier to express the functions in the form x = f(y) and one function is consistently to the right of the other, which is useful for curves like sideways parabolas (e.g., y² = 4ax).
8. What is the logic behind the formula ‘Area = ∫ [Upper Curve - Lower Curve] dx’?
This formula works by applying the fundamental concept of definite integrals. The integral of the upper curve, ∫ f(x) dx, calculates the entire area under it down to the x-axis. Similarly, the integral of the lower curve, ∫ g(x) dx, calculates the area under it. By subtracting the area under the lower curve from the area under the upper curve, we are left with precisely the area of the region enclosed between them.

















