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Area Under THE Curve formulas

Master Area Under THE Curve through 30 JEE Advanced-level formulas, systematically structured with every variable spelled out. Revise concept-wise, identify the areas where you need improvement, and focus your preparation with greater precision.

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Area Under THE Curve, every formula

30 formulas, typeset and free. Print it, or keep it open beside your practice.

Area under a curve (x-axis)
$A = \int_{a}^{b} f (x) d x, f (x) \geq 0 on [a, b]$
Area when f(x)≤0
$A = - \int_{a}^{b} f (x) d x = \int_{a}^{b} f (x) d x$
curve below x-axis
Area w.r.t. y-axis
$A = \int_{c}^{d} x d y = \int_{c}^{d} g (y) d y$
x=g(y)
Area between two curves
$A = \int_{a}^{b} [g (x) - f (x)] d x$
g(x)≥f(x) on [a,b]
Curve crossing the x-axis
$A = \int_{a}^{c} ∣ f ∣ d x + \int_{c}^{b} ∣ f ∣ d x$
split at the crossing x=c
Symmetry about x-axis
$even powers of y \Rightarrow symmetric about x-axis$
Q1MCQSymmetry x-axis
$y^{2}=4x$ is symmetric about:
- A
  the x-axis
- B
  the y-axis
- C
  the line $y=x$
- D
  the origin
Symmetry about y-axis
$even powers of x \Rightarrow symmetric about y-axis$
Q1MCQSymmetry y-axis
$y=x^{2}$ is symmetric about:
- A
  the y-axis
- B
  the x-axis
- C
  the line $y=x$
- D
  the origin
Symmetry about y=x
$eqn unchanged under x \leftrightarrow y \Rightarrow symmetric about y = x$
Q1MCQSymmetry y=x
$xy=1$ is symmetric about:
- A
  the line $y=x$
- B
  the x-axis
- C
  the y-axis
- D
  no line
Area of |x|+|y|=a
$A = 2 a^{2}$
Area of rhombus ax±by±c=0
$A = \frac{2 c ^{2}}{∣ ab ∣}$
Area bounded by √x+√y=√a
$A = \frac{a ^{2}}{6}$
with the coordinate axes
Q1MCQ√x+√y=√a
The area bounded by $\sqrt{x}+\sqrt{y}=\sqrt{a}$ and the axes is:
- A
  $\dfrac{a^{2}}{6}$
- B
  $\dfrac{a^{2}}{3}$
- C
  $\dfrac{a^{2}}{2}$
- D
  $2a^{2}$
Parabola y=ax²+bx+c & x-axis
$A = \frac{( b ^{2} - 4 a c ) ^{3/2}}{6 a ^{2}}$
Parabola & a line
$A = \frac{Δ ^{3/2}}{6 a ^{2}}$
Δ = discriminant of ax²+(b-m)x+(c-k)
Q1MCQParabola & line
The area between $y=ax^{2}+bx+c$ and $y=mx+k$ is:
- A
  $\dfrac{\Delta^{3/2}}{6a^{2}}$
- B
  $\dfrac{\Delta^{3/2}}{6a}$
- C
  $\dfrac{\Delta^{1/2}}{6a^{2}}$
- D
  $\dfrac{\Delta^{3}}{6a^{2}}$
Area of ellipse
$A = π ab$
x²/a²+y²/b²=1
Q1MCQEllipse area
The area of the ellipse $\dfrac{x^{2}}{9}+\dfrac{y^{2}}{4}=1$ is:
- A
  $6\pi$
- B
  $36\pi$
- C
  $13\pi$
- D
  $5\pi$
Area of circle
$A = π a^{2}$
x²+y²=a²
Q1MCQCircle area
The area of $x^{2}+y^{2}=16$ is:
- A
  $16\pi$
- B
  $4\pi$
- C
  $8\pi$
- D
  $256\pi$
Two parabolas y²=4ax, x²=4by
$A = \frac{16 ab}{3}$
Q1MCQTwo parabolas y²=4ax,x²=4by
The area between $y^{2}=4ax$ and $x^{2}=4by$ is:
- A
  $\dfrac{16ab}{3}$
- B
  $\dfrac{16a^{2}}{3}$
- C
  $\dfrac{8ab}{3}$
- D
  $\dfrac{ab}{3}$
Two parabolas y²=4ax, x²=4ay
$A = \frac{16 a ^{2}}{3}$
Q1MCQTwo parabolas equal a
The area between $y^{2}=4ax$ and $x^{2}=4ay$ is:
- A
  $\dfrac{16a^{2}}{3}$
- B
  $\dfrac{16a}{3}$
- C
  $\dfrac{8a^{2}}{3}$
- D
  $\dfrac{a^{2}}{3}$
Parabola y²=4ax & line y=mx
$A = \frac{8 a ^{2}}{3 m ^{3}}$
Q1MCQParabola & line y=mx
The area between $y^{2}=4ax$ and $y=mx$ is:
- A
  $\dfrac{8a^{2}}{3m^{3}}$
- B
  $\dfrac{8a^{2}}{3m}$
- C
  $\dfrac{8a}{3m^{3}}$
- D
  $\dfrac{a^{2}}{3m^{3}}$
Parabola & its latus rectum
$A = \frac{8 a ^{2}}{3}$
y²=4ax
Q1MCQParabola & latus rectum
The area between $y^{2}=4ax$ and its latus rectum is:
- A
  $\dfrac{8a^{2}}{3}$
- B
  $\dfrac{4a^{2}}{3}$
- C
  $\dfrac{16a^{2}}{3}$
- D
  $\dfrac{2a^{2}}{3}$
One arc of y=sin(ax)
$A = \frac{2}{a}$
with x-axis
y=sin(ax) over [0,nπ]
$A = \frac{2 n}{a}$
Q1MCQsin over [0,nπ]
The area of $y=|\sin x|$ over $[0,n\pi]$ (with $a=1$) is:
- A
  $2n$
- B
  $n$
- C
  $\dfrac{n}{2}$
- D
  $\pi n$
Astroid area
$A = \frac{3 π a ^{2}}{8}$
x^{2/3}+y^{2/3}=a^{2/3}
Q1MCQAstroid area
The area of the astroid $x^{2/3}+y^{2/3}=a^{2/3}$ is:
- A
  $\dfrac{3\pi a^{2}}{8}$
- B
  $\dfrac{3\pi a^{2}}{4}$
- C
  $\pi a^{2}$
- D
  $\dfrac{\pi a^{2}}{8}$
Cycloid arc area
$A = 3 π a^{2}$
x=a(θ-sinθ), y=a(1-cosθ)
Q1MCQCycloid arc
The area under one arch of the cycloid $x=a(\theta-\sin\theta),\ y=a(1-\cos\theta)$ is:
- A
  $3\pi a^{2}$
- B
  $\pi a^{2}$
- C
  $2\pi a^{2}$
- D
  $\dfrac{3\pi a^{2}}{2}$
Triangle: tangent, normal & x-axis
$A = \frac{1}{2} y_{1}^{2} m + \frac{1}{m}$
m = slope of tangent at P(x1,y1)
Q1MCQTangent-normal-x-axis triangle
The area of the triangle formed by the tangent, normal at $P(x_1,y_1)$ and the x-axis is:
- A
  $\tfrac12 y_1^{2}\left|m+\dfrac1m\right|$
- B
  $\tfrac12 x_1^{2}\left|m+\dfrac1m\right|$
- C
  $y_1^{2}|m|$
- D
  $\tfrac12 y_1 m$
Triangle from three lines
$A = \frac{1}{2} \sum \frac{( c _{1} - c _{2} ) ^{2}}{m _{1} - m _{2}}$
y=m_i x+c_i
Q1MCQTriangle from 3 lines
The area of the triangle formed by $y=m_i x+c_i$ is:
- A
  $\tfrac12\left|\sum\dfrac{(c_1-c_2)^{2}}{m_1-m_2}\right|$
- B
  $\tfrac12\left|\sum(c_1-c_2)^{2}\right|$
- C
  $\sum(m_1-m_2)$
- D
  $\tfrac12(c_1-c_2)$
Parallelogram from four lines
$A = \frac{( c _{1} - d _{1} ) ( c _{2} - d _{2} )}{a _{1} b _{2} - a _{2} b _{1}}$
Q1MCQParallelogram from 4 lines
The area of the parallelogram formed by the four lines is:
- A
  $\left|\dfrac{(c_1-d_1)(c_2-d_2)}{a_1b_2-a_2b_1}\right|$
- B
  $|c_1-d_1|$
- C
  $\dfrac{c_1 c_2}{a_1 a_2}$
- D
  $(a_1b_2-a_2b_1)$
Area = integral of top minus bottom
$A = \int_{a}^{b} (y_{top} - y_{bottom}) d x$
Sign: below-axis contributes negative
$\int_{a}^{b} f d x is net signed area, not total area$
Q1MCQSigned area
$\int_a^b f(x)\,dx$ where $f$ dips below the axis gives:
- A
  the net signed area
- B
  the total geometric area
- C
  always a positive number
- D
  zero
Parabolas y²=4a(x+a) & y²=4b(b-x)
$A = \frac{8}{3} ab (a + b)$
Q1MCQTwo shifted parabolas
The area bounded by $y^{2}=4a(x+a)$ and $y^{2}=4b(b-x)$ is:
- A
  $\dfrac{8}{3}\sqrt{ab}\,(a+b)$
- B
  $\dfrac{8ab}{3}$
- C
  $\dfrac{16ab}{3}$
- D
  $\dfrac{4}{3}\sqrt{ab}$
Astroid (x/a)^{2/3}+(y/b)^{2/3}=1
$A = \frac{3 π ab}{8}$
Q1MCQGeneral astroid
The area enclosed by $\left(\dfrac{x}{a}\right)^{2/3}+\left(\dfrac{y}{b}\right)^{2/3}=1$ is:
- A
  $\dfrac{3\pi ab}{8}$
- B
  $\dfrac{3\pi a^{2}}{8}$
- C
  $\pi ab$
- D
  $\dfrac{\pi ab}{8}$

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Area Under THE Curve formulas

Area Under THE Curve, every formula

Area under a curve (x-axis)

Area when f(x)≤0

Area w.r.t. y-axis

Area between two curves

Curve crossing the x-axis

Symmetry about x-axis

Symmetry about y-axis

Symmetry about y=x

Area of |x|+|y|=a

Area of rhombus ax±by±c=0

Area bounded by √x+√y=√a

Parabola y=ax²+bx+c & x-axis

Parabola & a line

Area of ellipse

Area of circle

Two parabolas y²=4ax, x²=4by

Two parabolas y²=4ax, x²=4ay

Parabola y²=4ax & line y=mx

Parabola & its latus rectum

One arc of y=sin(ax)

y=sin(ax) over [0,nπ]

Astroid area

Cycloid arc area

Triangle: tangent, normal & x-axis

Triangle from three lines

Parallelogram from four lines

Area = integral of top minus bottom

Sign: below-axis contributes negative

Parabolas y²=4a(x+a) & y²=4b(b-x)

Astroid (x/a)^{2/3}+(y/b)^{2/3}=1

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