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Area Under Curves formula sheet

Every formula in the Area Under Curves chapter for JEE Main, with one line on when it applies. Topic 1 is open to everyone; sign in free to read all 20 and download the PDF.

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How we count: 7× asked means 7 past JEE Main questions in the PrepWiser bank needed that formula, and "last" is the latest year one appeared. ★ marks formulas asked 8 or more times.

Area bounded by a curve and the axes

Area under a curve
6× asked, last 2025
A=∫ab∣f(x)∣ dxA=\int_a^b|f(x)|\,dx
Split at each root in (a,b)(a,b); every piece counts positive. y=x(3−x)y=x(3-x) on [−1,2][-1,2]: 11/6+10/3=31/611/6+10/3=31/6, not 3/23/2.
Exponential and log curves
12× asked, last 2026
∫ln⁡x dx=xln⁡x−x,∫ax dx=axln⁡a\int\ln x\,dx=x\ln x-x,\qquad \int a^x\,dx=\frac{a^x}{\ln a}
ln⁡x<0\ln x<0 on (0,1)(0,1): check which curve is on top. ∫12ln⁡y dy=2ln⁡2−1\int_1^2\ln y\,dy=2\ln2-1.
Circle-type integrals
10× asked, last 2026
∫a2−x2 dx=x2a2−x2+a22sin⁡−1xa∫x2−a2 dx=x2x2−a2−a22ln⁡∣x+x2−a2∣\begin{aligned}&\int\sqrt{a^2-x^2}\,dx=\frac x2\sqrt{a^2-x^2}+\frac{a^2}2\sin^{-1}\frac xa\\&\int\sqrt{x^2-a^2}\,dx=\frac x2\sqrt{x^2-a^2}\\&\qquad-\frac{a^2}2\ln\left|x+\sqrt{x^2-a^2}\right|\end{aligned}
Coefficient a2/2a^2/2; take the first term at both limits. ∫ab(b−x)(x−a) dx=π(b−a)2/8\int_a^b\sqrt{(b-x)(x-a)}\,dx=\pi(b-a)^2/8 (a semicircle).

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Practise Area Under Curves

17 more formulas in 4 topics

Area between two curves; Modulus, greatest integer and |x| + |y| regions; Standard areas: parabola, circle, ellipse; and Symmetry and areas with a parameter. Read the first part free. Sign in free with Google to read the rest and download the PDF.

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Area between two curves

  • Upper minus lower22×
  • Curves that cross6×
  • Horizontal strips21×
  • Regions given by inequalities18×
  • Min or max as a boundary16×

Modulus, greatest integer and |x| + |y| regions

  • Modulus curves23×
  • |x| + |y| ≤ k and rhombus4×
  • Greatest integer [x]3×

Standard areas: parabola, circle, ellipse

  • Parabola and a chord22×
  • Line through the vertex6×
  • Two parabolas, latus rectum4×
  • Parabolic segment7×
  • Ellipse and circle6×
  • Circle cut by a chord8×
  • Circle with a parabola11×

Symmetry and areas with a parameter

  • Symmetry20×
  • Finding a parameter14×

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About 140 past JEE Main questions from this chapter, with instant answer checking and step-by-step solutions where available. Free, and you can start without signing up.

Practise Area Under Curves

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