Areas Under Curves

Concept overview

Area between a curve y = f(x) and the x-axis over [a, b] is ∫_a^b |f(x)| dx. Absolute value handles the case where f dips below the x-axis — signed area vs geometric area matter.

Area between two curves y = f(x) and y = g(x) over [a, b], with f ≥ g, is ∫_a^b (f − g) dx. When one curve is above and the other below in different parts of the interval, split the integral at the intersection points and handle each region with the correct 'upper − lower'.

For regions best described in terms of y (say a curve x = h(y)), integrate with respect to y: ∫_c^d h(y) dy. Choosing the right variable of integration cuts problem time in half.

Common tricky configurations: area between a parabola and a chord, between a circle and a tangent line, or bounded by three curves creating a non-trivial region. Sketching the region first is non-negotiable.

Key formulas & concepts

• Area under f(x): A = ∫_a^b |f(x)| dx
• Area between f and g: A = ∫_a^b (f − g) dx where f ≥ g
• Area with respect to y: A = ∫_c^d h(y) dy
• Split at intersections: Break integral where curves cross
• Parabola vertical: y² = 4ax; area between y = ±2√(ax) and vertical chord

Common mistakes to avoid

• Forgetting the absolute value when the curve dips below the axis.
• Not splitting the integral at intersection points where the 'upper' and 'lower' curves swap.
• Choosing the harder variable of integration — always sketch the region first.
• Confusing signed area (integral) with geometric area (the physical region).

JEE Main relevance

JEE Main asks 1 area question every paper — usually a well-behaved region between a curve and a straight line or between two curves.

JEE Advanced relevance

Advanced problems build the region from parametric or polar-form curves, or involve areas that maximise/minimise with a parameter.

Previous-year questions (PYQ)

Parabola + line and circle + chord are the two most repeated area archetypes.