Area Under THE Curve formulas

Every formula JEE Advanced expects you to know for Area Under THE Curve, with the variables spelled out. Free, no sign-up. Read it before a mock, not during one.

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Formulas: 30
Subject: Mathematics
Exam: JEE Advanced
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Area Under THE Curve, every formula

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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
$even powers of y \Rightarrow symmetric about x-axis$
Symmetry about y-axis
$even powers of x \Rightarrow symmetric about y-axis$
Symmetry about y=x
$eqn unchanged under x \leftrightarrow y \Rightarrow symmetric about y = x$
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
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)
Area of ellipse
$A = π ab$
x²/a²+y²/b²=1
Area of circle
$A = π a^{2}$
x²+y²=a²
Two parabolas y²=4ax, x²=4by
$A = \frac{16 ab}{3}$
Two parabolas y²=4ax, x²=4ay
$A = \frac{16 a ^{2}}{3}$
Parabola y²=4ax & line y=mx
$A = \frac{8 a ^{2}}{3 m ^{3}}$
Parabola & its latus rectum
$A = \frac{8 a ^{2}}{3}$
y²=4ax
One arc of y=sin(ax)
$A = \frac{2}{a}$
with x-axis
y=sin(ax) over [0,nπ]
$A = \frac{2 n}{a}$
Astroid area
$A = \frac{3 π a ^{2}}{8}$
x^{2/3}+y^{2/3}=a^{2/3}
Cycloid arc area
$A = 3 π a^{2}$
x=a(θ-sinθ), y=a(1-cosθ)
Triangle: tangent, normal & x-axis
$A = \frac{1}{2} y_{1}^{2} m + \frac{1}{m}$
m = slope of tangent at P(x1,y1)
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
Parallelogram from four lines
$A = \frac{( c _{1} - d _{1} ) ( c _{2} - d _{2} )}{a _{1} b _{2} - a _{2} b _{1}}$
Area = integral of top minus bottom
$A = \int_{a}^{b} (y_{top} - y_{bottom}) d x$
Sign: below-axis contributes negative
$\int_{a}^{b} f d x is net signed area, not total area$
Parabolas y²=4a(x+a) & y²=4b(b-x)
$A = \frac{8}{3} ab (a + b)$
Astroid (x/a)^{2/3}+(y/b)^{2/3}=1
$A = \frac{3 π ab}{8}$

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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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