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

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

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