NCERT Solutions Class 12 Maths Chapter 8 teaches you how to apply integration to find areas under curves, between curves, and bounded regions. You’ll learn practical techniques to calculate areas using definite integrals, solve problems involving parabolas, circles, and ellipses, and master the step-by-step methods needed for board exam questions. These skills are essential for JEE preparation and understanding real-world applications of calculus.
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All exercises with step-by-step solutions | Updated 2025-26 | Free Download
Download PDF (Free)NCERT Solutions Class 12 Maths Chapter 8 Application of Integrals – Complete Guide
NCERT Class 12 Chapter 8 – Application of Integrals transforms your theoretical knowledge of integration into a powerful problem-solving tool. You’ll explore how definite integrals can calculate areas of complex geometric shapes, regions bounded by curves, and spaces between intersecting functions. This chapter bridges pure mathematics with practical applications, showing you why integration is fundamental in engineering, physics, and architecture.
📊 CBSE Class 12 Maths Chapter 8 – Exam Weightage & Marking Scheme
| CBSE Board Marks | 7 Marks |
| Unit Name | Calculus |
| Difficulty Level | Hard |
| Importance | High |
| Exam Types | CBSE Board, State Boards |
| Typical Questions | 2-3 questions |
The chapter focuses on two main applications: finding areas under curves and between curves, and calculating volumes of solids of revolution. You’ll work with parabolas, circles, ellipses, and straight lines, learning systematic approaches to set up and evaluate integrals for area calculations. The problems require you to visualize geometric regions, identify boundaries, choose appropriate integration limits, and apply the correct formulas – skills that are heavily tested in CBSE board examinations.
With a weightage of 7 marks and marked as high difficulty, this chapter typically appears as one 6-mark long answer question or a combination of shorter problems in your board exam. Questions often involve finding areas of regions bounded by two or more curves, requiring you to find intersection points, sketch graphs, and set up proper integral expressions. Strong performance here significantly impacts your overall mathematics score.
Quick Facts – Class 12 Chapter 8
| 📖 Chapter Number | Chapter 8 |
| 📚 Chapter Name | Application of Integrals |
| ✏️ Total Exercises | 2 Exercises |
| ❓ Total Questions | 9 Questions |
| 📅 Updated For | CBSE Session 2025-26 |
Mastering Application of Integrals requires solid understanding of Chapter 7 (Integrals) and coordinate geometry concepts. You’ll need to integrate functions confidently, work with absolute values when curves intersect, and interpret your answers geometrically. The CBSE marking scheme rewards clear diagrams, correct identification of limits, and step-by-step integration. Regular practice with NCERT solutions and previous year questions will help you tackle the challenging problems confidently and score maximum marks in this high-weightage chapter.
NCERT Solutions Class 12 Maths Chapter 8 – All Exercises PDF Download
Download exercise-wise NCERT Solutions PDFs for offline study
| Exercise No. | Topics Covered | Download PDF |
|---|---|---|
| Exercise 8.1 | Complete step-by-step solutions for 4 questions | 📥 Download PDF |
| Miscellaneous Exercise on Chapter 8 | Complete step-by-step solutions for 5 questions | 📥 Download PDF |
Application of Integrals – Key Formulas & Concepts
Quick reference for CBSE exams
| Formula | Description | When to Use |
|---|---|---|
| Area under a curve \\[A = \\int_{a}^{b} y \, dx\\] | Calculates the area bounded by the curve y = f(x), the x-axis, and the lines x = a and x = b. Note: Ensure y = f(x) is non-negative within the interval [a, b]. If the curve is below the x-axis, the integral will give a negative value, so take the absolute value. | Finding the area between a single curve and the x-axis within specified limits. |
| Area between a curve and y-axis \\[A = \\int_{c}^{d} x \, dy\\] | Calculates the area bounded by the curve x = g(y), the y-axis, and the lines y = c and y = d. Note: Ensure x = g(y) is non-negative within the interval [c, d]. If the curve is to the left of the y-axis, take the absolute value. | Finding the area between a single curve and the y-axis within specified limits. |
| Area between two curves (x-limits) \\[A = \\int_{a}^{b} |f(x) – g(x)| \, dx\\] | Calculates the area between two curves y = f(x) and y = g(x) from x = a to x = b. Note: Determine which function is ‘above’ the other within the interval [a, b]. If f(x) > g(x), then A = integral of (f(x) – g(x)). If they intersect, split the integral into multiple parts. | Finding the area enclosed between two curves where you have x-axis limits. |
| Area between two curves (y-limits) \\[A = \\int_{c}^{d} |h(y) – k(y)| \, dy\\] | Calculates the area between two curves x = h(y) and x = k(y) from y = c to y = d. Note: Determine which function is ‘to the right’ of the other within the interval [c, d]. If h(y) > k(y), then A = integral of (h(y) – k(y)). If they intersect, split the integral into multiple parts. | Finding the area enclosed between two curves where you have y-axis limits. |
| Area using integration \\[Area = \\int_{a}^{b} y \, dx = \\int_{a}^{b} f(x) \, dx\\] | The area of the region bounded by the curve y = f(x), the x-axis, and the lines x = a and x = b. Note: If area is below x-axis, take its absolute value. | Calculate area under a single curve with x-axis limits |
| Area of a circle (using integration) \\[Area = 4 \\int_{0}^{r} \\sqrt{r^2 – x^2} \, dx = \\pi r^2\\] | Calculates the area of a circle with radius r using integration. Note: Uses the equation of a circle x² + y² = r². Focus on the first quadrant and multiply by 4 due to symmetry. | Useful as a conceptual exercise. Not typically used for direct calculation, but helps demonstrate the application of integrals. |
| Area of an ellipse (using integration) \\[Area = 4 \\int_{0}^{a} \\frac{b}{a} \\sqrt{a^2 – x^2} \, dx = \\pi a b\\] | Calculates the area of an ellipse with semi-major axis ‘a’ and semi-minor axis ‘b’ using integration. Note: Uses the equation of an ellipse x²/a² + y²/b² = 1. Focus on the first quadrant and multiply by 4 due to symmetry. | Similar to circle, more of a conceptual application to understand integration, not for direct computation. |
| Finding Intersection Points \\[f(x) = g(x)\\] | To find the x-coordinates where two curves y=f(x) and y=g(x) intersect. Note: Solve the equation f(x) = g(x) to find the x values where the curves intersect. These x values will be your limits of integration. | Before finding area between two curves, you must find limits of integration by equating the functions. |
| Area of a region bounded by a parabola and a line \\[A = \\int_{a}^{b} (y_{line} – y_{parabola}) dx\\] | Calculates the area between a line and a parabola, where the limits of integration (a and b) are the x-coordinates of the intersection points. Note: First, find the points of intersection. Ensure you subtract the equation of the parabola from the equation of the line, if the line is above the parabola in the given interval. | When asked to find the area between a parabola (e.g., y² = 4ax) and a straight line (e.g., y = mx). |
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