CalculusHigh Yield (1-2 Qs in JEE)

Area Under Curves Formulas

Area bounded by curves, lines, parabolas, and standard shortcut formulas.

8 Formulas•346+ Questions in Bank•

Concept Diagram

Definite Integral & Tangent Line

Geometric interpretation of ∫ f(x) dx as area under curve and derivative f'(x₀) as tangent slope.

\text{Area} = \int_a^b [y_{\text{upper}}(x) - y_{\text{lower}}(x)] \, dx = \int_c^d [x_{\text{right}}(y) - x_{\text{left}}(y)] \, dy

Area enclosed between two intersecting curves.

Examiner Tip: Find points of intersection first to determine integration limits a and b.

\text{Area bounded by } y^2 = 4ax \text{ and } y = mx \implies \text{Area} = \frac{8a^2}{3m^3}

Direct closed-form formula saving 5 minutes of definite integration.

Examiner Tip: Area between y² = 4ax and x² = 4by is (16/3)ab.

\text{Area of } \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \implies \text{Area} = \pi a b

Total area enclosed by ellipse with semi-axes a and b.

Examiner Tip: Area of circle of radius r is special case: π r² (a = b = r).

\text{Area bounded by } y^2 = 4ax \text{ and } x^2 = 4by \implies \text{Area} = \frac{16}{3} a b

Legendary JEE shortcut for enclosed area between standard horizontal and vertical parabolas.

Examiner Tip: For y² = 4x and x² = 4y (a = 1, b = 1), the area is exactly 16/3 sq units.

\text{Area bounded by } y^2 = 4ax \text{ and } x = a \implies \text{Area} = \frac{8}{3} a^2

Area bounded by the parabola and the focal chord perpendicular to its principal axis.

Examiner Tip: Area bounded by y² = 4ax and double ordinate x = x₁ is (4/3) x₁ y₁.

\text{Loop of } a y^2 = x^2 (a - x) \implies \text{Area} = \frac{8}{15} a^2, \quad \text{Arch of Cycloid } = 3\pi a^2

Closed-form definite integration results for parametric and looped algebraic geometries.

Examiner Tip: Area of leaf of Folium of Descartes x³ + y³ = 3axy is (3/2) a².

\int_0^\pi \sin x\,dx = 2, \quad \text{Area between } \sin x \text{ and } \cos x \; \left[\frac{\pi}{4}, \frac{5\pi}{4}\right] = 2\sqrt{2}

Fundamental cycle areas for periodic trigonometric functions.

Examiner Tip: Area of a single full lobe between consecutive zero crossings of y = sin(kx) is 2/k.

f_{\text{avg}} = \frac{1}{b - a}\int_a^b f(x)\,dx, \quad f_{\text{rms}} = \sqrt{\frac{1}{b - a}\int_a^b [f(x)]^2\,dx}

Continuous integral definitions of the mean value and effective quadratic average over interval [a, b].

Examiner Tip: For alternating sinusoidal function f(t) = I₀ sin(ωt): f_avg over half-cycle is 2I₀/π ≈ 0.637 I₀; f_rms is I₀/√2 ≈ 0.707 I₀.

Frequently tested misconceptions in Area Under Curves

•Integrating across regions where curve dips below the x-axis: area is strictly non-negative, must take absolute value |f(x)|.
•Swapping upper and lower curves.