Class 12 Maths Chapter 8 Application of Integrals

Application of Integrals Class 12 introduces you to the geometrical applications of definite integrals, especially in finding areas of regions bounded by curves. You will learn how integration becomes a powerful tool for calculating areas of shapes that do not have straightforward geometric formulas. 

In real-life situations and competitive exams, areas bounded by functions, parabolas, exponentials, circles, and intersecting lines are extremely common. This chapter teaches you the skill to handle all types of curve-related area problems, making integration both practical and interesting.

Introduction to Applications of Integrals

The definite integral links algebra with geometry. When you compute the integral of a function y = f(x) from x = a to x = b, it represents the net signed area between the curve and the x-axis. This idea is the foundation of this entire chapter.

Irregular shapes do not have fixed formulas. For example:

• The area under y = x² does not have a direct geometrical formula
• The area bounded by a parabola and a line cannot be found using basic geometry
• The area enclosed between a sine curve and an axis changes continuously
  
  

In such cases, integration becomes the only method to calculate area with precision.

Geometrically:

Area = integral from a to b of f(x) dx

if the function remains above the axis. If it crosses the axis, careful splitting is needed.

Area Under Simple Curves

Consider a continuous function y = f(x) on the interval [a, b].

Basic Formula:
Area = integral from a to b of f(x) dx
when f(x) is non-negative throughout the interval.

However, many curves move above and below the x-axis. In such cases:

If the curve crosses the x-axis:

1. Identify the points where f(x) = 0
2. Break the interval into subintervals
3. Compute the integral separately on each interval
4. Take absolute value of each and add them
   

This ensures that the final answer represents physical area and not signed area.

Example of such a situation: The curve y = sin x between 0 and 2π changes sign at π. So you must split at π and compute separately.

Area Between a Curve and the y-axis

Sometimes expressions like x = g(y) are more natural. Examples include:

• A sideways parabola like x = y²
• Certain rotated curves
• Regions better understood vertically

In such cases, integration with respect to y is simpler.

Formula: Area = integral from y = c to y = d of g(y) dy

This approach gives clean solutions when vertical boundaries are easier to identify.

Area Between Two Curves

This is the most important part of the chapter and carries the highest weightage in board and competitive exams.

Suppose two curves y = f(x) and y = g(x) intersect at x = a and x = b.

If f(x) is the upper curve and g(x) is the lower curve:

Formula:
Area = integral from a to b of [f(x) − g(x)] dx

This subtracts the lower curve from the upper curve to get the vertical thickness of the strip.

Sometimes the curves are horizontal in nature and expressed as x = f(y) and x = g(y). Then:

Formula:
Area = integral from y = c to y = d of [f(y) − g(y)] dy

Integration with respect to y is required when:

• Horizontal thickness varies easily
• Curves bend sideways

Vertical integration would require splitting into too many regions

Procedure to Find Area Between Curves

1. Identify the curves
Write down their equations clearly and understand how they behave.

2. Sketch the region
A rough diagram (even if imperfect) is essential.
It helps identify:

• Which curve lies above or below
• Intersection points
• Orientation of the region
• Whether symmetry exists
  

3. Find intersection points
Solve the equations f(x) = g(x).
These values become your limits of integration.

4. Choose whether to integrate with respect to x or y
Ask yourself:

• Are vertical strips easier? Choose dx.
• Are horizontal strips easier? Choose dy.
  

5. Set up the integral
Write the integral as upper minus lower or right minus left.

6. Evaluate carefully
Apply integral rules, simplify, and compute precisely.

Important Notes for Students

• Always sketch the graph before solving
• Identify the correct upper and lower curves to avoid negative results
• Use symmetry whenever possible to reduce calculation
• Be careful with curves that change sign
• Confirm the limits using intersection points instead of guessing

These small steps improve accuracy significantly.

By now, you understand that definite integrals are far more than algebraic expressions. They form the core of geometric applications in mathematics. In this chapter, you have learned:

• How to find area under curves
• How to find area between curves
• How to handle curves defined as x = g(y)
• Why choosing the right variable for integration matters
• How sketching simplifies problem solving
• How integrals can solve real-life geometry problems
• How these concepts appear repeatedly in exams and competitive tests
  

Once you begin practicing, you will find that these problems become intuitive and even enjoyable. This chapter is powerful, practical and highly scoring.

FAQs

Q1. What is the main application of definite integrals studied in this chapter?
Ans. The primary application is calculating areas of regions bounded by curves, straight lines, the x-axis or y-axis. Definite integrals also help find the area between two curves by integrating the difference of their values.

Q2. How do we decide whether to integrate with respect to x or y?
Ans. The choice depends on the orientation of the region:

• If the region is easier to divide vertically, integrate with respect to x.
• If horizontal strips describe the region more naturally, integrate with respect to y.
  This choice makes the setup simpler and reduces mistakes.

Q3. What should we do if a curve lies partly above and partly below the x-axis?
Ans. You must locate the points where the curve crosses the axis, split the interval at those points, compute separate integrals for each interval, and take the absolute values to get the true physical area.

Q4. Why is sketching the curves important?
Ans. A sketch helps you identify the overall shape of the region, see which curve lies above or below, accurately locate intersection points and recognize symmetry. Without a sketch, mistakes in limits and curve ordering are more likely.

Q5. Is this chapter important for CBSE boards and competitive exams?
Ans. Yes. Applications of integrals carry significant weight in Class 12 Boards and appear frequently in competitive exams like JEE Main where questions on areas between curves are common. Mastering these methods gives a strong scoring advantage