Application of Integrals NCERT Class 12: Area and Formulas

Imagine trying to calculate the exact area of an irregular plot of land or the space under a curved graph—simple formulas like length × breadth won’t work here. This is where the Application of Integrals becomes powerful. It helps us measure areas bounded by curves, lines, and axes with precision. In this chapter, you’ll move beyond basic integration and learn how to apply it in real situations. By the end, you’ll be able to solve area-based problems confidently and score high in exams.

What is the Application of Integrals? 

In mathematics, integration is often introduced as the reverse process of differentiation. While differentiation helps us understand how quantities change, integration helps us accumulate or measure total quantities.

The Application of Integrals focuses on using this concept to find areas. Instead of just solving integrals as mathematical expressions, we now use them to solve real-world problems like:

• Finding the area under a curve
• Measuring land boundaries
• Calculating distance from velocity graphs 

Think of it this way:

Differentiation = Breaking things down
Integration = Adding things up

In Class 12, the main application you study is area calculation. When a curve is drawn on a graph, the space between the curve and the axis cannot be measured using basic geometry. Integration divides this space into infinitely small parts and adds them together to give the exact area.

Mathematically, if a function is given as , the area between the curve and the x-axis from to is:

This expression represents the total accumulated area.

In real life, engineers, architects, and scientists use integrals to measure irregular shapes, making this concept extremely practical and important.

Key Concept: Area Under a Curve

The most important idea in this chapter is understanding the area under a curve.

When a curve is drawn on a coordinate plane, we often need to calculate the area between:

• the curve
• the x-axis
• two vertical lines and

Unlike rectangles or triangles, this area is irregular. So how do we find it?

Conceptual Understanding

Integration works by dividing the entire area into infinitely thin vertical strips. Each strip has:

• Width = dx
• Height = f(x) 

So, area of one strip =

Adding all such strips from to , we get:

This gives the exact area, not an approximation.

Important Observations

1. If the curve lies above the x-axis, area is positive
2. If the curve lies below the x-axis, area is negative
3. For total area, we take absolute value 

Example

Find the area under the curve from 0 to 2

Solution:
=
= sq units

Why This Matters

This concept forms the base for all types of area problems:

• Between curves
• Between line and curve
• Complex bounded regions 

Once you understand this deeply, the rest of the chapter becomes much easier.

Area Bounded by Curve and X-axis 

This is the most basic application of integrals.

If a curve is given by , then the area between the curve and the x-axis from to is:

When to Use This Formula

Use this when:

• Function is in the form
• Limits are given in terms of x 

Special Case: Curve Crossing X-axis

If the curve crosses the x-axis:

• Area below x-axis becomes negative
• But area is always positive 

So we split the integral

Example idea:

Example

Find area under from 0 to π

Solution:
=
= 2 sq units

Key Tip

Always:

• Check sign of function
• Draw rough graph 

Even a small visualization saves marks in exams.

Area Between Two Curves 

This is one of the most important and frequently asked concepts in exams.

When two curves are given, the required area lies between them.

Formula

If upper curve =
and lower curve = , then:

Key Idea

Area = Upper curve – Lower curve

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Steps to Solve

1. Find points of intersection
2. Identify which curve is above
3. Apply formula
4. Solve integration 

Special Case: Curves Intersect

Sometimes curves cross each other:

• Upper curve changes
• Split the interval 

Example

Find area between and

Step 1:
x = x² → x = 0, 1

Step 2:
Upper curve = x
Lower curve = x²

Step 3:

Solution:
=
=

Pro Tip

• If confused, plug a value between limits
• The function giving larger value = upper curve 

Area with Respect to Y-axis 

Till now, you’ve mostly worked with integration using dx (with respect to x). But sometimes, problems become easier if we integrate with respect to y (dy) instead.

dx vs dy Concept

• When function is given as → use dx
• When function is given as → use dy 

Think of it like direction:

• dx → vertical strips
• dy → horizontal strips 

Formula (Area w.r.t Y-axis)

This represents the area between:

• curve
• y-axis  
• horizontal lines and

When to Use dy?

Use integration with respect to y when:

• The equation is easier in terms of x = f(y)
• The graph is sideways (like )
• Converting becomes complicated 

Example

Find the area bounded by , y = 0 to y = 2

Solution:
=
= sq units

Pro Tip

If you ever feel the integration is getting messy, try:
“Can I switch from dx to dy?”

This trick saves time and reduces calculation errors in exams.

Important NCERT Formulas

Here are the must-know formulas from NCERT that cover almost all questions:

Area Under a Curve

Used when curve is above x-axis

Area with Respect to Y-axis

Used when curve is in form

Area Between Two Curves

Upper curve – Lower curve

Area When Curves Intersect

If curves cross at point :

Split into two parts

Area Using Symmetry

If curve is symmetric about y-axis:

 Saves time in exams

Quick Tip

• Always confirm which curve is on top
• Always check if splitting is needed 

These formulas alone can solve 80% of board questions.

Step-by-Step Problem Solving Strategy — 250 words

Many students lose marks not because they don’t know formulas, but because they don’t follow a clear process.

Here’s a foolproof method:

Step 1: Understand the Question

• Identify what is given
• What area is required 

Step 2: Draw a Rough Sketch

This is the most important step

• Helps identify region
• Avoids confusion 

Step 3: Find Points of Intersection

• Solve equations
• These become limits 

Step 4: Decide Variable (dx or dy)

Ask yourself:

• Is it easier in x? → dx
• Is it easier in y? → dy 

Step 5: Identify Curves

• Upper curve / lower curve
  OR
• Right curve / left curve 

Step 6: Apply Formula

• Substitute limits
• Use correct expression 

Step 7: Solve Carefully

• Avoid algebra mistakes
• Keep steps clean 

Golden Rule

“Graph + Limits + Correct Formula = Full Marks”

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Common Mistakes Students Make

Even strong students lose marks due to small mistakes. Here are the most common ones:

Wrong Limits

Students often:

• Forget to find intersection points
• Use incorrect values 

Always solve equations properly

Not Checking Sign

• Area below the x-axis becomes negative
• Final answer should be positive 

Use absolute value if needed

Confusing Upper and Lower Curve

• Taking wrong function as upper
• Leads to negative answer 

Plug a value to check

Not Drawing Graph

Skipping graph = confusion

Even a rough sketch helps

Using dx Instead of dy

• Makes integration complicated
• Increases chances of error 

Final Tip

Most mistakes are avoidable with:

• Proper visualization
• Step-by-step approach 

Also Read

Explore 330+ NCERT Solutions – Free to Download

Practice Questions: Area Under Curve

Solve the following to strengthen your basics:

1. Find the area under from x = 0 to x = 3
2. Find the area under from x = 1 to x = 4
3. Find the area under from x = 0 to x = π
4. Find the area under from x = 0 to x = 1
5. Find the area under from x = 0 to x = 2
6. Find the area under from x = 0 to x = π/2
7. Find the area under from x = 1 to x = 2
8. Find the area under from x = 0 to x = 4
9. Find the area under from x = 0 to x = 2
10. Find the total area under from x = –2 to x = 2

Practice Questions: Area Between Two Curves

Solve the following:

1. Find the area between and
2. Find the area between and
3. Find the area between and
4. Find the area between and
5. Find the area between and x-axis from 0 to π
6. Find the area between and
7. Find the area between and
8. Find the area between and from 0 to 1
9. Find the area between and x-axis from –2 to 2  
10. Find the area between and

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Practice Questions: Curve with Y-axis 

1. Find area bounded by , y = 0 to 2
2. Find area bounded by , y = 1 to 3
3. Find area bounded by , y = 0 to 1
4. Find area bounded by , y = 0 to 4
5. Find area between , y = 0 to 2
6. Find area between and x = 4
7. Find area between and
8. Find area between and y-axis
9. Find area between , y = 0 to 2
10. Find area between , y = 0 to 3 

Mixed Application Questions (Exam Level)

1. Find area between parabola and line
2. Find area between circle and x-axis
3. Find area enclosed by ellipse
4. Find area between and (0 to π/2)
5. Find area bounded by , y = 4 and x-axis
6. Find area enclosed by triangle using integration
7. Find area between and x-axis from 0 to 2
8. Find area between and line y = x
9. Find area between curve and axis using symmetry
10. Find area bounded by three curves 

NCERT-Based Important Questions 

(From your NCERT reference)

1. ,
2. ,
3. , , x = 0
4. ,
5. ,
6. ,
7. , x = 2
8. Area under line
9. from 0 to 1
10. Line , x-axis, x = 2 to 8 

Case-Based / Word Problems 

1. Find area of a curved land boundary given by
2. Calculate water tank surface area (parabolic shape)
3. Find area under speed-time graph (distance)
4. Estimate area of a bridge arch (semi-circle)
5. Find area of garden boundary between two curves
6. Calculate solar panel exposure area (curve model)
7. Find area under river flow curve
8. Area of a gate shaped like parabola
9. Area between road curves (traffic design)
10. Area between cost and revenue curves 

Solved Examples (Detailed)

Example 1: Area Under Curve

Find area under from 0 to 2

Solution:

Step 1: Apply integration
=

Step 2: Substitute limits
=

Final Answer = 8/3 sq units

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Example 2: Area Between Two Curves

Find area between and

Solution:

Step 1: Intersection
x = x² → x = 0, 1

Step 2: Identify curves
Upper = x, Lower = x²

Step 3: Integrate
=

Step 4: Solve
=

Final Answer = 1/6 sq units

Example 3: Area Using Y-axis

Find area bounded by , y = 0 to 2

Solution:

=

Final Answer = 8/3 sq units

Example 4: Area Under Trigonometric Curve

Find area under , 0 to π

Solution:

=
= 2

Final Answer = 2 sq units

Example 5: Area of Circle (Using Integration Idea)

For circle

Area = πa²

Derived using integration by splitting into halves

Most Expected Exam Questions

Based on NCERT patterns and board trends, these types of questions are highly expected:

1. Area Between Line and Parabola

• Very common
• Requires intersection + subtraction 

2. Area Between Two Curves

• Focus on identifying upper/lower curve
• Often 4–5 marks 

3. Trigonometric Area

• ,
• Limits usually 0 to π or π/2 

4. Circle / Ellipse Area

• Direct formula or integration-based 

5. Symmetry-Based Questions

• Saves time
• Reduces calculation 

6. Split Region Problems

• Curve crosses axis
• Requires breaking into parts 

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Final Strategy

Practice these patterns:

• 2 questions per type
• Focus on graph understanding 

If you master:

• Limits  
• Curve identification
• Formula selection 

You can easily score 90%+ in this chapter

Why Learn Application of Integrals with PlanetSpark?

Mastering topics like Application of Integrals isn’t just about memorising formulas—it’s about understanding concepts, building confidence, and applying them in real situations. That’s exactly where PlanetSpark comes in.

At PlanetSpark, learning goes beyond textbooks. Students are taught through interactive sessions, real-life examples, and personalized attention, making complex topics like integration simple and engaging.

What Makes PlanetSpark Different?

• Concept Clarity First
  Instead of rote learning, students understand why formulas work. This is crucial for topics like area under curves and bounded regions.
• 1:1 Live Classes
  Personalized guidance ensures every doubt is solved—especially helpful for tricky calculus problems.
• Application-Based Learning
  Students practice real-world problems (like area, graphs, and interpretations), making maths more relatable.
• Confidence & Communication Skills
  Beyond academics, PlanetSpark also focuses on helping students express their understanding clearly, which is important in exams and beyond.
• Expert Mentors
  Trained teachers simplify difficult topics into easy steps—just like we did in this guide. 

From Confusion to Control: Wrapping It Up

This chapter isn’t about tough maths—it’s about clarity. Once you start seeing curves as areas instead of equations, everything clicks. Most problems follow the same pattern: draw, decide, apply. That’s it. The real difference comes from practice and avoiding silly mistakes. Stay consistent, trust the process, and this chapter can easily become your scoring advantage in exams.

Also Read 

Learn Vectors Step by Step for Class 12 Maths