Class 12 Maths Chapter 8 – Applications Of Intergrals NCERT Solutions – FREE PDF Download | Hand – Written Notes

Chapter 8 Applications of Integrals explains how integration is used to find the area of regions bounded by curves and straight lines. While Chapter 7 teaches how to integrate, this chapter teaches where integration is applied practically.

This chapter is very important for:

• Board examinations
• Geometry-based numerical questions
• Understanding area problems in calculus

At Edu Tehri, this chapter is explained with diagrams, step-by-step methods, clear concepts, handwritten notes, and free PDFs, so students can visualize and solve problems easily.

What Are Applications of Integrals?

In simple words, applications of integrals mean using integration to calculate areas.

Earlier, students studied areas of:

• Squares
• Rectangles
• Circles

But in calculus, regions can be bounded by:

• Curves
• Straight lines
• Axes

To find the area of such regions, integration is used.

Main Idea of Chapter 8

This chapter is based on one main concept:

Area = ∫ (Upper curve − Lower curve) dx

or

Area = ∫ (Right curve − Left curve) dy

Understanding this idea is the key to mastering Chapter 8.

Area Under Simple Curves

Area Between a Curve and X-Axis

If a curve y = f(x) lies above the x-axis between x = a and x = b, then:

Area = a∫b f(x) dx

This is the most basic application of integrals.

Area When Curve Lies Below X-Axis

If the curve lies below the x-axis, the integral gives a negative value, but area is always positive.

So, we take absolute value of the integral.

Area Between Two Curves (Very Important)

This is the most important topic of Chapter 8.

Case 1: Area Between Two Curves (w.r.t x)

If:

• Upper curve = y = f(x)
• Lower curve = y = g(x)

Then area is given by:

Area = a∫b [f(x) − g(x)] dx

Case 2: Area Between Two Curves (w.r.t y)

If:

• Right curve = x = f(y)
• Left curve = x = g(y)

Then area is:

Area = ∫ [f(y) − g(y)] dy

Important Step

• Draw a rough sketch
• Identify upper/lower or left/right curve

Area Between a Curve and a Line

Many NCERT problems include:

• One curve
• One straight line

Steps to solve:

1. Find points of intersection
2. Decide upper and lower curve
3. Apply integration formula

Area Between a Curve and Coordinate Axes

Some regions are bounded by:

• Curve
• x-axis
• y-axis

In such cases:

• Decide limits carefully
• Use definite integrals

These questions are common in 3–4 mark board questions.

Graphical Understanding (Concept Booster)

Although exact graphs are not drawn in exams, rough sketches are very helpful.

Benefits of sketching:

• Helps identify limits
• Avoids mistakes
• Makes solution clearer

At Edu Tehri, every area problem is explained with a clear diagram-based approach.

Important Formulas – Chapter 8

Download Important Formula - Free PDF

Basic Area Formula

1. Area under curve:

A = a∫b f(x) dx

2. Area Between Two Curves (w.r.t x)

A = a∫b [Upper curve − Lower curve] dx

3. Area Between Two Curves (w.r.t y)

A = ∫ [Right curve − Left curve] dy

Important Points

• Area is always positive
• Limits come from intersection points
• Sketch before solving

Key Features of Chapter 8

• Application-based chapter
• Geometry + calculus combined
• Scoring if concepts are clear
• Requires diagram understanding
• Direct NCERT-based questions

Frequently Asked Questions

Q1. What is the main use of Chapter 8?

It is used to find the area of regions bounded by curves and lines.

Q2. Why is sketch important in this chapter?

Sketch helps identify limits and correct curves.

Q3. Is Chapter 8 difficult?

No, it becomes easy with diagram practice.

Q4. Are NCERT questions enough for boards?

Yes, board questions are mostly NCERT-based.

Classs- 12th Mathematics NCERT Solutions All Chapters - Hand Written Notes | Free download PDF's | 2026-2027