Chapter 8: Application of Integrals Class 12 Solutions PDF | NCERT Maths Handwritten Notes Download

Chapter 8: Application of Integrals Class 12 Solutions PDF | NCERT Maths Handwritten Notes Download: Chapter 8, Application of Integrals, is one of the most important and interesting chapters in Class 12 Mathematics. This chapter focuses on how definite integrals can be applied in real-life mathematical problems, such as calculating the area under curves and between two curves. It is a practical extension of what you’ve learned in the chapter on Integration. The chapter introduces students to the concept of finding the area bounded by a curve, lines, and axes in the coordinate plane using integration.

The NCERT solutions provided here are detailed, step-by-step, and fully based on the latest CBSE syllabus. These solutions help in understanding various techniques and formulas involved in calculating area, especially when dealing with curves represented by algebraic functions. These notes also include graphical representations to help visualize the bounded areas clearly.

Our handwritten notes PDF offers concise summaries, formulas, solved NCERT questions, and extra practice problems, making it easier for students to revise and score better in board exams. Whether you’re a quick learner or need extra help, these solutions are crafted to support all types of learners. Download the free PDF and master the concepts of this chapter effectively.

Preview of Chapter 8: Application of integrals Class 12 Math’s Solutions | NCERT | Free Handwritten Notes Download PDF

---

Images of Chapter 8 Maths Class 12 Solutions PDF | NCERT | Free Handwritten Notes Download PDF

Related Free Notes from Studycart24.com of Class 12 Math’s

• Class 12 Computer Science Notes PDF in English Free Download
• Class 12 Geography Notes: Download Free PDF of All Chapters in English
• Complete Class 12 Chemistry Notes PDF with All Chapters for Free
• Class 12 Physics Handwritten Notes in English PDF Download
• Class 12 Biology Handwritten Notes PDF Free Download- StudyCart24.com
• Class 12 Business Studies Notes: FREE Notes Chapter 1 to 12

---

Suggested Notes Links from Amazon Related to Class 12 Math’s Handwritten Notes

---

Key Features of Application of integrals Class 12 Solutions Math’s | NCERT | Chapter 8 Maths Class 12 Solutions PDF

• Subject: Maths (Chapter 8: Application of integrals class 12 Solutions)
• Language : Hindi
• Total pages :4
• File size: 1.3 MB
• Format : PDF
• Well structured and easy to understand
• Includes importance formulas and definitions
• Covers all NCERT syllabus topics
• Useful for quick revision before exam

---

Definition of Chapter 8: Application of Integrals – Class 12 Math’s

The chapter Application of Integrals in Class 12 Mathematics deals with the practical use of definite integrals in calculating areas bounded by curves, lines, and coordinate axes. It is a real-world application of integration learned in previous chapters. The main objective is to teach students how to find the area under a curve or the area between two curves using integration with proper limits. This method is especially useful when dealing with irregular shapes that cannot be solved using simple geometry.

The chapter covers two major topics:

1. Area under a curve with respect to one of the coordinate axes.
2. Area between two given curves.

By using definite integrals and understanding the geometry of graphs, students learn to visualize and calculate these areas precisely. This chapter builds a strong foundation for further studies in engineering, physics, and other fields where integration is widely used.

Important Definitions Chapter 8: Application of Integrals – Class 12 Maths

Definite Integral:
A definite integral represents the area under a curve between two specified limits. It is written as: ∫abf(x) dx\int_a^b f(x) \, dx∫ab​f(x)dx

where aaa and bbb are the limits, and f(x)f(x)f(x) is the function.

Area Under a Curve:
It refers to the region bounded between a given curve y=f(x)y = f(x)y=f(x), the x-axis, and vertical lines x=ax = ax=a and x=bx = bx=b. Calculated using definite integration.

Area Between Two Curves:
If two curves y=f(x)y = f(x)y=f(x) and y=g(x)y = g(x)y=g(x) intersect between x=ax = ax=a and x=bx = bx=b, the area between them is: ∫ab[f(x)−g(x)] dx\int_a^b [f(x) – g(x)] \, dx∫ab​[f(x)−g(x)]dx

where f(x)≥g(x)f(x) \geq g(x)f(x)≥g(x) in the given interval.

Bounded Region:
A closed area enclosed by curves or lines in the coordinate plane, within which integration is applied to find the total area.

Integrable Function:
A function that can be integrated over a given interval is called integrable. It must be continuous or have a finite number of discontinuities.

---

How to Prepare for Chapter 8: Application of Integrals – Class 12 Maths

---

✅ 1. Understand the Basics of Integration First

Before starting this chapter, revise definite integrals from Chapter 7. Understanding the concept of limits and areas is crucial here.

✅ 2. Learn Key Formulas Thoroughly

Memorize and understand the basic formulas for:

• Area under a curve: Area=∫abf(x) dx\text{Area} = \int_a^b f(x)\, dxArea=∫ab​f(x)dx
• Area between two curves: Area=∫ab[f(x)−g(x)] dx\text{Area} = \int_a^b [f(x) – g(x)]\, dxArea=∫ab​[f(x)−g(x)]dx

✅ 3. Practice NCERT Examples and Exercises

Solve all NCERT examples and exercises. Pay special attention to questions where:

• One function is a line and the other is a parabola
• Curves intersect and you need to find the points of intersection

✅ 4. Use Graphs to Visualize

Sketching the graphs of functions helps understand which area you are integrating, especially between two curves.

✅ 5. Solve Previous Year Questions (PYQs)

Practice questions from previous board exams to know the common patterns and frequently asked problems.

✅ 6. Attempt Extra Problems from Reference Books

Try additional questions from books like RS Aggarwal or RD Sharma for more confidence.

✅ 7. Focus on Units and Accuracy

Write correct units for area (usually square units) and avoid calculation errors, especially in limits and subtraction.

---

Chapters / Subtopics Included in Chapter 8: Application of Integrals – Class 12 Math’s

📘 1. Introduction

A brief overview of how definite integrals are used to calculate areas under curves and between curves.

---

📘 2. Area Under Simple Curves

• Area bounded by a curve y=f(x)y = f(x)y=f(x), the x-axis, and vertical lines x=ax = ax=a and x=bx = bx=b
• Use of definite integrals to find such areas
• Graphical representation and visualization

---

📘 3. Area Between Two Curves

• Area enclosed between two curves y=f(x)y = f(x)y=f(x) and y=g(x)y = g(x)y=g(x)
• Formula: Area=∫ab[f(x)−g(x)] dx\text{Area} = \int_a^b [f(x) – g(x)]\, dxArea=∫ab​[f(x)−g(x)]dx
• Finding points of intersection of curves
• Using symmetry wherever applicable

---

📘 4. Solved Examples (NCERT)

Illustrative examples demonstrating:

• Area under parabola, circle, line, etc.
• Area between two intersecting curves
• Application of limits and absolute values

---

📘 5. Miscellaneous Exercise

Mixed problems involving all concepts
Useful for board-level practice and revision

---

Why Are These Handwritten Notes Special for You? – Chapter 8: Application of Integrals – Class 12 Math’s

✍️ 1. Easy-to-Understand Language

The notes explain tough concepts like area under curves and between curves in simple, student-friendly language.

📌 2. Step-by-Step NCERT Solutions

Each example and exercise question from the NCERT textbook is solved step-by-step, making it easier to follow the logic and method.

🧠 3. Concept Highlights & Formula Box

Important formulas and key concepts are highlighted in boxes, perfect for quick revision before exams.

🔍 4. Neat Diagrams and Graphs

Hand-drawn graphs help visualize the region under or between curves, improving conceptual clarity.

📖 5. Board-Focused Presentation

The notes follow CBSE exam patterns, including past year questions and commonly asked application-based problems.

✅ 6. Time-Saving Revision Tool

Since everything is neatly compiled, you save time during last-minute revision. No need to flip through bulky books!

---

Top 10 Benefits of Using Handwritten Notes – Chapter 8: Application of Integrals – Class 12 Math’s

✅ 1. Concept Clarity

Handwritten notes explain every concept—like area under curves—in a clear, step-by-step manner.

✅ 2. Neatly Structured Content

The notes are organized in a proper flow: definitions → formulas → examples → exercises.

✅ 3. Visual Learning through Graphs

Graphs and diagrams help visualize the bounded areas, making understanding easier.

✅ 4. Quick Revision Friendly

Important points, formulas, and shortcuts are highlighted—perfect for fast last-minute revisions.

✅ 5. Easy to Memorize

Writing patterns and underlined keywords help students retain formulas and steps longer.

✅ 6. CBSE Exam-Oriented

Includes past year questions and frequently asked problems that align with the CBSE pattern.

✅ 7. Time-Saving Resource

No need to refer to multiple books—everything is compiled in one concise format.

✅ 8. Improves Writing Practice

Reading handwritten solutions helps you improve your own presentation in board exams.

✅ 9. Error-Free Solutions

Prepared by experts or toppers, so the steps are accurate and exam-relevant.

✅ 10. Boosts Confidence

When you’re well-prepared with simple and effective notes, your confidence in solving integration problems increases.

---

Important Topics of Chapter 8: Application of Integrals – Class 12 Maths

S. No.	Topic Name	Description
1.	Area Under a Curve	Area between a curve y=f(x)y = f(x)y=f(x), the x-axis, and vertical lines x=ax = ax=a and x=bx = bx=b.
2.	Area Between Two Curves	Area between y=f(x)y = f(x)y=f(x) and y=g(x)y = g(x)y=g(x) using definite integrals.
3.	Use of Symmetry	Applying symmetry about x-axis or y-axis to simplify calculations.
4.	Finding Points of Intersection	Solving f(x)=g(x)f(x) = g(x)f(x)=g(x) to find limits of integration.
5.	Graphical Representation	Drawing curves and bounded regions to visualize the required area.
6.	Application of Integration in Geometry	Calculating areas bounded by lines, parabolas, circles, etc.
7.	Questions Based on Real-life Shapes	Solving area-related problems based on standard curves like x2x^2×2, x\sqrt{x}x​, etc.
8.	NCERT Exercise and Miscellaneous Questions	Practicing all types of questions from the NCERT book thoroughly.

---

📘 Class 12 Math’s Important Formulas – Chapter 8: Application of Integrals

S. No.	Formula	Description
1.	Area=∫abf(x) dx\text{Area} = \int_a^b f(x)\, dxArea=∫ab​f(x)dx	Area under the curve y=f(x)y = f(x)y=f(x)
2.	( \text{Area} = \int_a^b	f(x)
3.	Area=∫ab[f(x)−g(x)] dx\text{Area} = \int_a^b [f(x) – g(x)]\, dxArea=∫ab​[f(x)−g(x)]dx	Area between two curves y=f(x)y = f(x)y=f(x) and y=g(x)y = g(x)y=g(x), where f(x)≥g(x)f(x) \ge g(x)f(x)≥g(x)
4.	Area bounded by curves=∫x1x2f(x) dx\text{Area bounded by curves} = \int_{x_1}^{x_2} f(x)\, dxArea bounded by curves=∫x1​x2​​f(x)dx	Used when region is between vertical lines
5.	Area between curves=∫y1y2[x2−x1] dy\text{Area between curves} = \int_{y_1}^{y_2} [x_2 – x_1]\, dyArea between curves=∫y1​y2​​[x2​−x1​]dy	Area when curves are expressed as x=f(y)x = f(y)x=f(y)
6.	Total Area=∫acf(x) dx+∫cbg(x) dx\text{Total Area} = \int_a^c f(x)\, dx + \int_c^b g(x)\, dxTotal Area=∫ac​f(x)dx+∫cb​g(x)dx	When different functions define area over different intervals
7.	If curve symmetric about y-axis: 2∫0af(x) dx\text{If curve symmetric about y-axis: } 2 \int_0^a f(x)\, dxIf curve symmetric about y-axis: 2∫0a​f(x)dx	For symmetric curves (like even functions)
8.	If curve symmetric about x-axis: 2∫0af(y) dy\text{If curve symmetric about x-axis: } 2 \int_0^a f(y)\, dyIf curve symmetric about x-axis: 2∫0a​f(y)dy	For x-axis symmetric curves
9.	Area of a circle: A=πr2A = \pi r^2A=πr2	Used for problems involving circle segments
10.	Area of semi-circle: A=12πr2A = \frac{1}{2} \pi r^2A=21​πr2	For curves like y=r2−x2y = \sqrt{r^2 – x^2}y=r2−x2​

---

📚 FAQs on Application of Integrals Class 12 Solutions PDF

---

✅ Class 12 Math’s Preparation Tips – Chapter 8: Application of Integrals

🔢	Tip
1.	Revise Integration Basics before starting – especially definite integrals.
2.	Understand the concept of area under curves and between two curves visually.
3.	Memorize key formulas and practice applying them directly.
4.	Draw graphs for every problem—this helps in identifying limits and upper/lower curves.
5.	Solve all NCERT examples and exercises step-by-step.
6.	Focus on questions with intersecting curves—they are frequent in exams.
7.	Use symmetry in curves (about x-axis/y-axis) to simplify area calculations.
8.	Avoid skipping graph questions—they clarify your understanding and fetch marks.
9.	Practice previous year board questions (PYQs) on this chapter.
10.	Do timed practice of mixed questions from miscellaneous exercises.

---

⚠️ Avoid These 15 Common Mistakes in Class 12 Math’s

🔢	Common Mistake	Tip to Avoid It
1.	Forgetting to apply limits in definite integration	Always write limits clearly after integrating.
2.	Missing modulus sign when area lies below x-axis	Use absolute value or split integral accordingly.
3.	Confusing upper and lower functions in area between two curves	Sketch graph to identify which curve is on top.
4.	Wrong limits of integration from graphs or intersections	Solve equations to find correct points of intersection.
5.	Using indefinite integration formulas instead of definite ones	Practice both types; use definite integral formulas in this chapter.
6.	Ignoring unit of area in answers	Always write “sq. units” or appropriate units in final answer.
7.	Poor graph plotting skills	Practice standard curve graphs (parabola, circle, line, etc.).
8.	Calculation mistakes in subtraction or definite integral evaluation	Double-check calculations step-by-step.
9.	Not breaking integration into parts when function changes in interval	Use separate integrals when curve behavior changes.
10.	Applying formulas without understanding concepts	Focus on visualizing the area being calculated.
11.	Leaving constants of integration in definite integrals	No “+C” in definite integrals—only use in indefinite.
12.	Incorrect substitution method in integrals	Verify substitution and revert if needed.
13.	Skipping questions with diagrams in the exam	These often carry easy marks if graphs are correct.
14.	Not checking domain and range of functions before integration	Make sure curves are valid over the given interval.
15.	Relying only on rote learning of formulas without practice	Regular practice is key to mastering Maths.

---

📘 Summary Table – Chapter 8: Application of Integrals (Class 12 Math’s)

Topic	Key Concept
Area under a Curve	Use definite integral: ∫abf(x) dx\int_a^b f(x) \, dx∫ab​f(x)dx
Area Between Two Curves	Use formula: ∫ab[f(x)−g(x)] dx\int_a^b [f(x) – g(x)] \, dx∫ab​[f(x)−g(x)]dx, where f(x)≥g(x)f(x) \geq g(x)f(x)≥g(x)
Use of Modulus	When function lies below x-axis: ( \int_a^b
Limits of Integration	Found using points of intersection or given boundary conditions
Symmetry in Curves	Use symmetry to simplify: 2∫0af(x) dx2 \int_0^a f(x) \, dx2∫0a​f(x)dx if curve is symmetric about y-axis
Graphical Understanding	Essential for identifying upper/lower functions and correct limits
Important Curves	Standard curves: Parabola y=x2y = x^2y=x2, Line y=mx+cy = mx + cy=mx+c, Circle x2+y2=r2x^2 + y^2 = r^2×2+y2=r2
Units of Area	Final answer must be in square units (e.g., sq. units or sq. cm)
NCERT Exercises	Includes example problems, exercise questions, and a miscellaneous section
Application Skills	Useful in real-life scenarios like physics, engineering, and geometry

---

✅ Conclusion – Application of Integrals Class 12 Solutions PDF (Class 12 Math’s)

Chapter 8 – Application of Integrals is one of the most practical and visual topics in Class 12 Mathematics. It extends your understanding of definite integrals by applying them to calculate the area under curves and between two curves. This chapter builds a strong link between algebraic concepts and geometric interpretation, helping you visualize areas using graphs and solve problems using integration techniques.

Through this chapter, you learn to handle real-world mathematical problems—such as calculating the area of irregular shapes that cannot be solved using standard geometry formulas. Key topics include finding the area under simple curves, between intersecting curves, and using symmetry to simplify problems. The use of definite integrals in these contexts not only strengthens your calculus skills but also prepares you for competitive exams and higher studies in fields like physics, engineering, and data analysis.

Mastering this chapter requires regular practice, clear concept understanding, and familiarity with graph-based problems.

Tags:-
Class 12 Maths Application of Integrals Notes
Application of Integrals Class 12 NCERT Solutions
Application of Integrals Class 12 Handwritten Notes PDF
Chapter 8 Maths Class 12 Important Formulas
Class 12 Maths Area Under Curve Solutions
NCERT Solutions Chapter 8 Class 12 Maths
Application of Integrals Class 12 CBSE Board Preparation
Free PDF Download Class 12 Maths Chapter 8
Application of Integrals Class 12 Summary
Application of Integrals Class 12 Practice Questions
CBSE Class 12 Maths Chapter 8 Short Notes
Application of Integrals Important Questions with Solutions
Class 12 Maths Formulas PDF Chapter 8
Application of Integrals Class 12 Graph-Based Questions
Board Exam Preparation Class 12 Maths Chapter 8