Math · Calculus

Applications of the Integrals formulas for JEE

Every Applications of the Integrals formula you need for JEE, grouped by concept.

14 formulas1 concepts

14 formulas · tap a card to reveal1 / 14

QuestionState the formula — Area Below x-axis

Reveal answer

Area Below x-axis

A = \left| \int_{a}^{b} f(x) \, dx \right|

Area bounded by a curve that lies entirely below the x-axis.applies whenf(x) < 0 on [a, b]areamodulusbelow_axis

↺ Tap to flip back · next card below

All 14 Applications of the Integrals formulas

Area under and between Curves

14 formulas

Area Below x-axis

A = \left| \int_{a}^{b} f(x) \, dx \right|

Area bounded by a curve that lies entirely below the x-axis.

applies whenf(x) < 0 on [a, b]

areamodulusbelow_axis

Area of a Circle

A = \pi a^2

Total area enclosed by the circle x^2 + y^2 = a^2.

applies whenDerived via

4 \int_{0}^{a} y \, dx

areacirclestandard

Elementary Vertical Strip Area

dA = y \, dx

Area of an infinitesimally thin vertical strip of height y and width dx.

applies wheny = f(x)

areaelementary_strip

Area of an Ellipse

A = \pi ab

Total area enclosed by the ellipse x^2/a^2 + y^2/b^2 = 1.

applies whenDerived via

4 \int_{0}^{a} y \, dx

areaellipsestandard

Area Under Curve (Horizontal Strips)

A = \int_{c}^{d} g(y) \, dy

Total area bounded by the curve x = g(y), y-axis, and horizontal lines y = c and y = d.

applies whenCurve is to the right of y-axis; g(y) >= 0 on [c, d]

areahorizontal

Fundamental Identity of Inverse Function Areas

\int_{a}^{b} f(x) \, dx + \int_{f(a)}^{f(b)} f^{-1}(y) \, dy = b f(b) - a f(a)

Relates the area under a function and the area under its inverse.

applies whenf(x) is strictly monotonic and continuous.

areainverse_functionjee-advanced

Modulus Function Area Expansion

y = x|x| \implies y = \begin{cases} x^2, & x > 0 \\ -x^2, & x < 0 \end{cases}

Piecewise expansion required to integrate functions involving absolute values multiplied by variables.

applies whenMust be evaluated piecewise around roots of the modulus term.

areamoduluspiecewise

Area of Parametric Curves

A = \left| \int_{t_1}^{t_2} y(t) x'(t) \, dt \right|

Area bounded by a curve defined parametrically as x=x(t), y=y(t) and the x-axis.

applies whenCurve traced once as t goes from t_1 to t_2.

areaparametricjee-advanced

Area in Polar Coordinates

A = \frac{1}{2} \int_{\alpha}^{\beta} r^2 \, d\theta

Area bounded by a polar curve r = f(\theta) and the radial lines \theta = \alpha, \theta = \beta.

applies whenSectorial area from the origin.

areapolarjee-advanced

Area of Curve Crossing x-axis

A = \left| \int_{a}^{c} f(x) \, dx \right| + \int_{c}^{b} f(x) \, dx

Total area when a curve crosses the x-axis at x = c, having portions both above and below the axis.

applies whenc is a root where f(c) = 0

arearootssplitting

Standard Integral for Circular/Elliptical Areas

\int \sqrt{a^2 - x^2} \, dx = \frac{x}{2}\sqrt{a^2 - x^2} + \frac{a^2}{2}\sin^{-1}\left(\frac{x}{a}\right) + C

Crucial standard integration formula used to find the areas of circles and ellipses.

applies when-a <= x <= a

integrationstandard_formulaconics

Area Between Two Curves

A = \int_{a}^{b} |f(x) - g(x)| \, dx

Total bounded area between two continuous functions, automatically handling crossing points.

applies whenIntersection points must be found to remove the absolute value signs for actual evaluation.

areatwo_curvesjee-advanced

Area Bounded by Two Standard Parabolas

A = \frac{16ab}{3}

Standard result for the area enclosed between parabolas y^2 = 4ax and x^2 = 4by.

applies whena > 0, b > 0

areaparabolasshortcutjee-advanced

Area Under Curve (Vertical Strips)

A = \int_{a}^{b} f(x) \, dx

Total area bounded by the curve y = f(x), x-axis, and ordinates x = a and x = b.

applies whenCurve is above x-axis; f(x) >= 0 on [a, b]

areavertical

Rhovecs schedules these formulas back to you right before you’d forget them — and picks the next concept to practise. We decide, you execute.

Get started