Area under Curves & Limit as Sums

Introduction

• The process of finding the area of a plane region is called quadrature.
• Area bounded by curves is evaluated using definite integrals.
• A rough sketch of the curve is essential before setting up the integral.

Curve Tracing — Important Checks

1. Symmetry

• Symmetric about x-axis if all powers of y are even
• Symmetric about y-axis if all powers of x are even
• Symmetric about origin if f(−x, −y) = f(x, y)
• Examples:
• y² = 4ax → x-axis symmetry
• x² = 4ay → y-axis symmetry
• y = x³ → origin symmetry

2. Origin

• If the equation has no constant term, the curve passes through the origin
• Example: x² + y² + 2ax = 0

3. Intercepts with Axes

• Put y = 0 to find x-intercepts
• Put x = 0 to find y-intercepts
• Example: x²/a² + y²/b² = 1 cuts axes at (±a, 0), (0, ±b)

4. Region of the Curve

• Express the equation as y = f(x)
• Find the interval of x for which y is real
• Example: For xy² = a²(a − x), y is real for 0 < x ≤ a

Area Bounded by a Curve

A. Area w.r.t. x-axis

• If curve is y = f(x) between x = a and x = b:
• Area = ∫_a^b f(x) dx

B. Area w.r.t. y-axis

• If curve is x = f(y) between y = c and y = d:
• Area = ∫_c^d f(y) dy

C. Parametric Form

• If x = f(t), y = g(t):
• Area = ∫ y dx = ∫ g(t) f′(t) dt

Symmetrical Area

• If the curve is symmetrical, find area of one part and multiply
• Reduces calculation significantly

Positive & Negative Area

• Area is always taken as positive
• If region lies partly above and below x-axis:
• Calculate areas separately and add their absolute values

Area Between Two Curves

Case I: Curves intersect at two points

• If curves y = f₁(x) and y = f₂(x) intersect at x = a and x = b:
• Area = ∫_a^b [f₁(x) − f₂(x)] dx
• Upper curve − lower curve

Case II: Area bounded with x-axis

• If curves meet x-axis at different points:
• Area = ∫_a^c f₁(x) dx + ∫_c^b f₂(x) dx

Limit as Sums

• Used to find limits of infinite sums using definite integrals
• Key formula:
• lim_n→∞ (1/n) Σ f(r/n) = ∫₀¹ f(x) dx

Method to Evaluate Limit as Sum

• Write rth (or r−1th) term in the form f(r/n)
• Factor out 1/n
• Replace:
• Σ → ∫
• r/n → x
• Lower limit: r = 0 → x = 0
• Upper limit: r = n−1 → x = 1

JEE Main Focus Tips

• Always sketch curves before integrating
• Check symmetry to reduce effort
• Identify upper and lower curves correctly
• Limit as sum problems are direct but concept-heavy
• Convert sums carefully into integral form