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Area Under Curves revision notes

Every idea in the Area Under Curves chapter for JEE Main, with the usual mistakes and the fast methods. 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 concept, and "last" is the latest year one appeared. ★ marks ideas 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
Region between y=f(x)y=f(x), the xx-axis and x=ax=a, x=bx=b. Split at each root of ff in (a,b)(a,b); every piece counts positive.
xxyyO23−1-111/611/610/310/3
y=x(3−x)y=x(3-x) on [−1,2][-1,2]: area 31/631/6, signed integral 3/23/2
  • Not ∫abf dx\int_a^b f\,dx when ff changes sign. The part below the axis cancels. y=x(3−x)y=x(3-x) on [−1,2][-1,2]: pieces 11/611/6 and 10/310/3 give area 31/631/6; the signed integral is only 3/23/2.
Exponential and log curves
12× asked, last 2026
∫ln⁡x dx=xln⁡x−x,∫ax dx=axln⁡a,∫dxx+k=ln⁡∣x+k∣\int \ln x\,dx = x\ln x - x,\qquad \int a^{x}\,dx=\frac{a^{x}}{\ln a},\qquad \int\frac{dx}{x+k}=\ln|x+k|
a>0a>0, a≠1a\ne1. ln⁡x<0\ln x<0 on (0,1)(0,1) and ex>0e^x>0 always: check which curve is on top before subtracting.
xxyy122ln⁡2−12\ln2-1ln⁡x<0\ln x<0 on (0,1)(0,1)y=ln⁡xy=\ln x
∫12ln⁡x dx=[xln⁡x−x]12=2ln⁡2−1\int_1^2\ln x\,dx=[x\ln x-x]_1^2=2\ln2-1; dropping −x-x gives 2ln⁡22\ln2
  • Not yln⁡yy\ln y alone for ∫ln⁡y dy\int\ln y\,dy. The −y-y term is easy to drop and changes the answer by the interval length: ∫12ln⁡y dy=2ln⁡2−1\int_1^2\ln y\,dy=2\ln2-1, not 2ln⁡22\ln2.
Circle-type integrals
10× asked, last 2026
∫a2−x2 dx=x2a2−x2+a22sin⁡−1xa∫x2−a2 dx=x2x2−a2−a22ln⁡∣x+x2−a2∣\begin{gathered}\int\sqrt{a^2-x^2}\,dx=\frac{x}{2}\sqrt{a^2-x^2}+\frac{a^2}{2}\sin^{-1}\frac{x}{a}\\ \int\sqrt{x^2-a^2}\,dx=\frac{x}{2}\sqrt{x^2-a^2}-\frac{a^2}{2}\ln\left|x+\sqrt{x^2-a^2}\right|\end{gathered}
First for circles (∣x∣≤a|x|\le a), second under a hyperbola branch (∣x∣≥a|x|\ge a). Same forms in yy.
xxyyO1x2+y2=5x^2+y^2=5y2=4xy^2=4x(1,±2)(1,\pm2)
Inside both: 2/3+5sin⁡−1(2/5)2/3+5\sin^{-1}(2/\sqrt5)
  • Not (a/2)sin⁡−1(a/2)\sin^{-1}. The coefficient is a2/2a^2/2, and the (x/2)a2−x2(x/2)\sqrt{a^2-x^2} term must be taken at both limits. Circle x2+y2=5x^2+y^2=5 inside y2≤4xy^2\le4x: 2/3+5sin⁡−1(2/5)2/3+5\sin^{-1}(2/\sqrt5).
  • Quick method. Spot circles instead of integrating: ∫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); An odd part like t1−t2t\sqrt{1-t^2} on [−1,1][-1,1] gives 0.

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Practise Area Under Curves

17 more concepts in 5 topics

Area between two curves; Modulus, greatest integer and |x| + |y| regions; Standard areas: parabola, circle, ellipse; and Symmetry and areas with a parameter, plus ideas from other chapters. 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 lower (vertical strips)22×
  • Curves that cross6×
  • Horizontal strips (right minus left)21×
  • Regions given by inequalities18×
  • Min or max as a boundary16×

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

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

Standard areas: parabola, circle, ellipse

  • Parabola and a chord22×
  • Parabola and a line through its vertex6×
  • Two parabolas, latus rectum, power curves4×
  • Parabolic segment (Archimedes)7×
  • Ellipse and circle6×
  • Circle cut by a chord8×
  • Circle or ellipse with a parabola11×

Symmetry and areas with a parameter

  • Symmetry: compute one part, then double20×
  • Finding a parameter from an area14×

Also asked here: other chapters

  • Key points1

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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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