Area Using Definite Integrals: Under Curves and Between Curves

• Compute the area enclosed between a curve and the x-axis (handling regions below x-axis)
• Find the area between two curves by integrating their difference
• Locate intersection points of curves to set up correct limits
• Apply standard area formulas for circle, ellipse, and parabola
• Distinguish signed area from geometric (unsigned) area

Level 1 — Foundations

The definite integral ∫ₐᵇ f(x)dx gives the signed area: positive when f(x) > 0 (above x-axis), negative when f(x) < 0 (below x-axis).

For geometric area (always positive), use absolute value: $A = \int_a^b |f(x)|\,dx$

When the curve crosses the x-axis at c ∈ (a,b), split: $A = \int_a^c f(x)\,dx + \int_c^b (-f(x))\,dx = \int_a^c f(x)\,dx - \int_c^b f(x)\,dx$

Area between two curves y = f(x) (upper) and y = g(x) (lower) from x = a to x = b: $A = \int_a^b [f(x) - g(x)]\,dx \quad \text{(when } f(x) \geq g(x) \text{ on } [a,b])$

For geometric area when curves switch position: $A = \int_a^b |f(x) - g(x)|\,dx$

Level 2 — JEE depth

Area under curve y=f(x) from a to b (below x-axis): $A = \int_a^b |f(x)|\,dx = -\int_a^b f(x)\,dx \quad [\text{if } f(x) \leq 0 \text{ on } [a,b]]$

Area between y=f(x) and y=g(x):

1. Find intersection: solve f(x) = g(x) → limits a, b
2. Determine which is on top in [a,b]
3. $A = \int_a^b |f(x)-g(x)|\,dx$

Standard results:

Area of circle by integration: $A = 4\int_0^r \sqrt{r^2-x^2}\,dx = 4\cdot\frac{\pi r^2}{4} = \pi r^2$

Area bounded by parabola y²=4ax and line x=a: At x=a: y=±2a (intersection points) $A = 2\int_0^a 2\sqrt{ax}\,dx = 4\sqrt{a}\cdot\frac{2}{3}a^{3/2} = \frac{8a^2}{3}$

Area using y as variable: When curves are expressed as x=f(y), integrate w.r.t. y: $A = \int_c^d |f(y) - g(y)|\,dy$

Signed vs geometric area:

• Signed: ∫ₐᵇ f(x)dx (may be negative)
• Geometric: ∫ₐᵇ |f(x)|dx (always non-negative)
• JEE always asks for geometric area unless explicitly stated otherwise

Find the area enclosed between y = x² and y = x + 2.

Step 1: Find intersection points
  x² = x + 2  →  x² − x − 2 = 0  →  (x−2)(x+1) = 0
  x = −1, x = 2

Step 2: Determine which is on top
  At x = 0: x+2 = 2 > 0 = x²  → line is above parabola on [−1, 2]

Step 3: Set up integral
  A = ∫₋₁² [(x+2) − x²] dx
    = ∫₋₁² (x + 2 − x²) dx

Step 4: Evaluate
  = [x²/2 + 2x − x³/3]₋₁²
  = (4/2 + 4 − 8/3) − (1/2 − 2 + 1/3)
  = (2 + 4 − 8/3) − (1/2 − 2 + 1/3)
  = (6 − 8/3) − (−3/2 + 1/3)

  Compute each:
  6 − 8/3 = 18/3 − 8/3 = 10/3
  −3/2 + 1/3 = −9/6 + 2/6 = −7/6

  A = 10/3 − (−7/6) = 10/3 + 7/6 = 20/6 + 7/6 = 27/6 = 9/2

Area = 9/2 square units

• Find the area between the parabola y=x² and the line y=2x
• Find the area of the region bounded by y²=4x and x=4
• Find the area enclosed between y=sin x and y=cos x from 0 to π/2
• Find the area of the ellipse x²/9 + y²/4 = 1
• Sketch and find the area of the region bounded by y=|x| and y=2

• Fig 8.1: Area under a curve as the integral
• Fig 8.3: Area between two curves (upper minus lower)
• Fig 8.7: Region bounded by parabola and line — standard sketch
• Sketch any circle/parabola/ellipse by hand before integrating; marks are often given for correct sketch

• Area under y=x² from 0 to 3: → [x³/3]₀³ = 9
• Area between y=x and y=x² from 0 to 1: → ∫₀¹(x−x²)dx = 1/2−1/3 = 1/6
• Area of upper semicircle x²+y²=4: → (π·4)/2 = 2π
• If the integral gives a negative value for an area problem, what should you do? → Take its absolute value (it means the curve was below the axis)
• Stretch: Find the area common to the circle x²+y²=4 and the parabola y²=x (set up the integral; leave in integral form if needed).

NCERT Chapter 8 link: Application of Integrals — full chapter (only chapter) with Exercises 8.1 and 8.2 and Miscellaneous
Exam connections: Directly tested in JEE Main (1–2 questions per paper); requires solid integration skill from Chapter 7; connects to coordinate geometry (circles, parabolas, ellipses from Class 11)
Study strategy: Sketching is non-negotiable — 60% of errors come from not drawing the region. Draw every area problem before integrating. Master 5 standard curve pairs: line+parabola, parabola+parabola, circle+line, ellipse.

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• Stretch: "How does this connect to coding, robotics, money, health, environment, or a future career?"

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• One hands-on project or measurement using the Drishti kit or household items that makes the concept physical.
• Direct link to at least one Future Skill track (Money Management, Green Tech, Cyber Defenders, Micro-Entrepreneurship, AI Mastery, Sustainable Living, Personality Development).
• Coding extension where relevant (simple script, simulation, or data logging).

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