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MathsMediumClass 12

Area Between Two Curves

Definite Integration & Area

17

JEE Qs

8%

Hard

75

min

Always sketch the given curves accurately to visualize the bounded region, identify all intersection points, and correctly set up the definite integral(s).

🧮 Key Formulas

Area = integral_a^b |f(x) - g(x)| dx
If f(x) >= g(x) on [a, b], Area = integral_a^b (f(x) - g(x)) dx
If f(y) >= g(y) on [c, d], Area = integral_c^d (f(y) - g(y)) dy

✅ Key Points for JEE

  • 1Always sketch the graphs of the given curves to visualize the region and identify all intersection points, which define the limits of integration.
  • 2Correctly determine which function is the 'upper' curve and which is the 'lower' curve (or 'right' and 'left' for integration with respect to y) in each sub-region.
  • 3If the 'upper' and 'lower' curves swap positions within the integration interval, the integral must be split into multiple parts, using the absolute value property, to ensure the area is always positive.
  • 4Choose whether to integrate with respect to x (dx) or y (dy) based on which orientation simplifies the setup of the integrand and the limits of integration.

⚠️ Common Mistakes

  • ✕Incorrectly identifying the upper/lower (or right/left) function, leading to a negative area or an incorrect magnitude.
  • ✕Failing to find all intersection points between the curves, which can result in missing parts of the total area or incorrect limits.
  • ✕Not splitting the integral when the curves intersect and cross each other, thereby miscalculating the total area due to cancellation of positive and negative signed areas.

📝 Practice Questions

See all

Q1. Let f(x) = ∫t0 (1) 253 (2) 154 (3) 125 (4) 157 →

2025·NumericalHard

Q11.Let the area enclosed between the curves |y| = 1 −x2 and x2 + y2 = 1 be α. If 9α = βπ + γ; β, γ are integers, then the value of |β −γ| equals. (1) 27 (2) 33 (3) 15 (4) 18

2025·MCQMedium

Q21.If 24 ∫ 0 4 (sin 4x − 12π + [2 sin x])dx = 2π + α, where [⋅] denotes the greatest integer function, then α is equal to _______.

2025·NumericalMedium

Q6. Let for f(x) = 7 tan8 x + 7 tan6 x −3 tan4 x −3 tan2 x, I1 = ∫π/40 f(x)dx and I2 = ∫π/40 xf(x)dx. Then 7I1 + 12I2 is equal to : (1) 2 (2) 1 (3) 2π (4) π

2025·MCQMedium

Q13.The area of the region, inside the circle (x −2√3)2 + y2 = 12 and outside the parabola y2 = 2√3x is : (1) 3π + 8 (2) 6π −16 (3) 3π −8 (4) 6π −8

2025·MCQHard

Q7. The area of the region enclosed by the curves y = x2 −4x + 4 and y2 = 16 −8x is : (1) 8 (2) 4 3 3 (3) 8 (4) 5 x ∈R. Then the numbers of local maximum and local minimum points of f ,

2025·MCQMedium

NCERT Chapters

  • Class 12 Maths Ch 8: Application of Integrals