Area Between Curves: JEE Main Integration Applications

Area between curves is one of JEE Main's most visual chapters, contributing two to three marks every session. The key skill is not the integration itself (which is usually elementary) but setting up the integral correctly: identifying which function is on top, finding the intersection points, and deciding whether to integrate with respect to x or y. Students who can draw a quick sketch of the region before integrating make far fewer setup errors than those who go algebraically from the start.

Area Under a Curve: The Basic Setup

The area bounded by y = f(x), the x-axis, and vertical lines x = a, x = b is ∫_a^b |f(x)| dx. The absolute value is critical — when f(x) is negative in part of [a, b], the signed integral gives a net value, not the geometric area. Split the interval at the zeros of f to compute total area correctly: if f changes sign at x = c in [a, b], total area = ∫_a^c f(x)dx − ∫_c^b f(x)dx (adjusting signs so each portion is positive).

The area between two curves y = f(x) and y = g(x) from x = a to x = b, where f(x) ≥ g(x) throughout, is ∫_a^b [f(x) − g(x)] dx. If the curves cross within [a, b], split at the crossing point and apply the formula separately in each sub-interval, always subtracting the lower curve from the upper. Finding the crossing points is usually the first computational step — set f(x) = g(x) and solve. Take a free area-applications mock to practice setting up these integrals.

Standard Curve Families and Their Areas

The most common curves in JEE area problems: parabola y² = 4ax (area between the parabola and chord x = a is (4/3)a² — a result worth memorising directly); ellipse x²/a² + y²/b² = 1 (area = πab); circle x² + y² = r² (area = πr²). These closed-form areas let you solve problems without explicit integration. When the region is formed by two parabolas, a parabola and a line, or a circle and a chord, finding intersection points and applying the standard formula almost always works.

Symmetry exploitation: many JEE area problems have a symmetric region about the x-axis, y-axis, or origin. For a region symmetric about the x-axis, compute the area above the x-axis and double it. This eliminates the need to handle negative y values and simplifies the computation substantially. The parabola-and-chord area formula emerges this way: the region between y² = 4ax and x = a is symmetric about the x-axis, so area = 2∫_0^{4a} √(4ax) dx, which gives (8/3)a² × (1/2) = ... the final result is (2/3) × base × height of the parabolic segment.

Integration with Respect to y

When the region is better described as bounded by x = f(y) curves rather than y = f(x) curves (horizontal slicing instead of vertical), integrate with respect to y: area = ∫_c^d [x_right(y) − x_left(y)] dy. This approach is necessary when the curve is not a function of x in the usual sense (e.g., for a parabola x = y² − 4, which is a sideways parabola). Recognise this setup by the word "horizontal strip" in the problem, or by noticing that the boundary curves are expressed more naturally as x in terms of y.

A combined strategy: for a region bounded by y = f(x) on one side and x = g(y) on another, sometimes integrating partly in x and partly in y, or rotating the axes conceptually, is more efficient than any single approach. The ability to choose the correct orientation is what separates fast solvers from slow ones on area questions. Connect this chapter to our definite integration guide for the evaluation techniques, and to our parabola guide for the curve families.