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Cheat sheet · Class 12 · CBSE · Science Mathematics

Application of Integrals cheat sheet

The whole chapter on one page: key formulas, facts and the points to remember, for quick revision before a test.

CBSE Class 12 · Mathematics

Cheat sheet

Chapter 8Application of Integrals

Area = ∫ (upper curve − lower curve) dx.

1. Area Under a Curve

A=∫aby dxA = \int_a^b y\,dxA=∫cdx dyA = \int_c^d x\,dy

If the curve is below the x-axis, take the absolute value.

2. Standard Results

  • Circle x2+y2=a2x^2 + y^2 = a^2: area πa2\pi a^2.
  • Ellipse x2a2+y2b2=1\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1: area πab\pi ab.

3. Method

  • Sketch the region.
  • Find intersection points (limits).
  • Choose vertical or horizontal strips.
  • Integrate and use symmetry.

Area of y2=4xy^2 = 4x bounded by x=1x = 1: 2∫012x dx=832\int_0^1 2\sqrt x\,dx = \dfrac83.

4. Key Points

  • Area is always positive.
  • Use symmetry to halve the work.
  • Answer in square units.
  • Check which curve is on top.

Remember: ∫ y dx | circle & ellipse areas | sketch first

Application of Integrals: questions and notes

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