📐 Mathematics — Class XII · Calculus

Application of Integrals

Finding areas bounded by curves — where integration becomes geometry

📖 Chapter 8 ⏱ ~30 min read 🏷 Calculus

In this chapter

Introduction
Area under Simple Curves
Summary

8.1 Introduction

"One should study Mathematics because it is only through Mathematics that nature can be conceived in harmonious form." — Birkhoff

Elementary geometry gives us formulae for areas of triangles, rectangles, and circles. But what about the area under a parabola, or between two curves? Integral calculus provides the tools to calculate these areas precisely.

8.2 Area under Simple Curves

The area under a curve can be thought of as the sum of infinitely many thin vertical (or horizontal) strips.

Figure 8.1 — Vertical strips (integrate w.r.t. x) vs horizontal strips (integrate w.r.t. y)

📐 Two Fundamental Formulae

Vertical strips (w.r.t. x):
Area = ∫ₐᵇ y dx = ∫ₐᵇ f(x) dx

Horizontal strips (w.r.t. y):
Area = ∫cᵈ x dy = ∫cᵈ g(y) dy

Area Between Two Curves

Figure 8.2 — Subtract the lower curve from the upper curve in the region of interest

Area Between Curves A = \intₐᵇ [f(x) - g(x)] dx where f(x) \geq g(x) on [a, b]

⚠️ Important Notes

(i) If the curve is below the x-axis, take absolute value: |∫ₐᵇ f(x) dx|
(ii) When curves cross, split the integral at intersection points
(iii) Use symmetry: area of circle = 4 × (area in first quadrant)

Area of Circle and Ellipse

Figure 8.3 — Circle area = πa², Ellipse area = πab — derived using integration

8.3 Summary

✅ Key Takeaways

• Vertical strips: Area = ∫ₐᵇ f(x) dx — integrate w.r.t. x

• Horizontal strips: Area = ∫cᵈ g(y) dy — integrate w.r.t. y

• Between curves: A = ∫ₐᵇ [f(x) − g(x)] dx (upper − lower)

• Signed area: area below x-axis → take absolute value

• Circle: A = πa² · Ellipse: A = πab

• Use symmetry to simplify calculations when possible

Ch 7 — Integrals Ch 9 — Differential Equations