Learning Objectives
- Calculate area under curves using definite integrals
- Find area between two curves
- Apply integration to find areas bounded by lines and conics
Key Concepts
Area Under a Curve
The area bounded by curve y = f(x), x-axis, and vertical lines x = a and x = b: A = ∫ab |f(x)| dx. If f(x) ≥ 0 on [a,b]: A = ∫ab f(x)dx. If f(x) ≤ 0 on [a,b]: A = -∫ab f(x)dx = ∫ab |f(x)|dx. If the curve crosses x-axis, split the integral at the crossing points.
Similarly, area bounded by x = g(y), y-axis, y = c, y = d: A = ∫cd |g(y)| dy.
Area Between Two Curves
Area between y = f(x) (upper curve) and y = g(x) (lower curve) from x = a to x = b: A = ∫ab [f(x) - g(x)] dx when f(x) ≥ g(x) on [a,b]. If curves intersect within the interval, find intersection points and split the integral.
Standard Areas (JEE Important)
Circle x2 + y2 = r2: Area = πr2. Calculate using 4∫0r √(r2-x2)dx.
Ellipse x2/a2 + y2/b2 = 1: Area = πab. Calculate using 4∫0a (b/a)√(a2-x2)dx.
Parabola y2 = 4ax: Area bounded by parabola and latus rectum (x = a): A = (4/3)a × 2a = 8a2/3.
Triangle with vertices: Determine equations of sides, set up integrals between appropriate limits.
Steps to Find Area
(1) Sketch the curves. (2) Find intersection points. (3) Identify upper and lower curves. (4) Set up the integral(s). (5) Evaluate. Always take absolute value or split integral when curves cross.
Summary
Definite integrals calculate areas under curves and between curves. The key is to correctly identify limits, upper/lower curves, and split integrals at crossing points. Standard areas of circles, ellipses, and parabolas can be derived using integration.
Important Terms
- Area under curve: ∫ₐᵇ |f(x)|dx between curve and x-axis
- Area between curves: ∫ₐᵇ [f(x) - g(x)]dx where f ≥ g
- Intersection points: Where curves meet; determine integration limits
Quick Revision
- Area under y = f(x) above x-axis: ∫ₐᵇ f(x)dx
- Area between curves: ∫ₐᵇ (upper - lower)dx
- Area of circle = πr2; Ellipse = πab
- Split integral if curves intersect in between
- When curve is below x-axis, take absolute value
- For symmetric curves, use symmetry to simplify (e.g., 2× or 4×)
- Draw sketch first to identify the bounded region