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MathsHardClass 12

Area Under / Between Curves

Definite Integration & Area

86

JEE Qs

28%

Hard

120

min

Always sketch the given curves precisely, identify all intersection points, and correctly determine the upper and lower functions (or right and left functions) before setting up the integral(s).

🧮 Key Formulas

Area = Integral from a to b of y dx = Integral from a to b of f(x) dx
Area = Integral from c to d of x dy = Integral from c to d of g(y) dy
Area between curves = Integral from a to b of |f(x) - g(x)| dx (where f(x) and g(x) are functions of x)
Area between curves = Integral from c to d of |g(y) - h(y)| dy (where g(y) and h(y) are functions of y)
Area in parametric form (x=phi(t), y=psi(t)) = Integral from t1 to t2 of y * (dx/dt) dt

✅ Key Points for JEE

  • 1Always begin by accurately sketching all given curves to correctly visualize the region and identify intersection points.
  • 2Determine the limits of integration by finding the points of intersection between the curves or from the given boundaries.
  • 3Decide whether to integrate with respect to 'x' (dx) or 'y' (dy); choose the variable that simplifies the expression and avoids splitting the integral unnecessarily.
  • 4The area between two curves f(x) and g(x) is given by Integral of (Upper Curve - Lower Curve) dx. If the upper/lower curve changes, split the integral at intersection points.
  • 5Utilize symmetry whenever possible to reduce calculation, e.g., if a region is symmetric about the x-axis or y-axis, calculate for one half and multiply by two.

⚠️ Common Mistakes

  • ✕Incorrectly sketching curves, leading to wrong limits of integration or wrong identification of upper/lower functions.
  • ✕Failure to find all intersection points between curves, resulting in incomplete area calculation.
  • ✕Ignoring the absolute value or taking (lower curve - upper curve) instead of (upper curve - lower curve), leading to negative or incorrect area values.
  • ✕Choosing the wrong variable of integration (dx vs dy) when one would be significantly simpler than the other.

📝 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 Mathematics Ch 7: Integrals
  • Class 12 Mathematics Ch 8: Application of Integrals