Arc LengthJEE Advanced

Measuring curves with √(1 + y'²) — interactive Mathematics simulation for IIT-JEE.

Concept

Length of a curve = limit of inscribed polylines. Each tiny piece is a hypotenuse: $ds = \sqrt{dx^2 + dy^2} = \sqrt{1 + (y')^2}\,dx$ . Integrate ds and the polyline converges to the true arc length — Pythagoras, infinitely often.

Key formula

L = \int_a^b \sqrt{1 + \left(\frac{dy}{dx}\right)^2}\,dx; \qquad \text{parametric: } L = \int\sqrt{\dot x^2 + \dot y^2}\,dt

Derivation

Chord over [x, x+Δx]: $\sqrt{\Delta x^2 + \Delta y^2} = \Delta x\sqrt{1 + (\Delta y/\Delta x)^2}$ . Let Δx → 0: the difference quotient becomes y′ and the sum becomes the integral.

Few curves have elementary arc lengths — y = x^{3/2}, the catenary cosh (its length is sinh!), and circles. Parabolas and sine already need special functions; that's WHY numeric quadrature matters.

Scenarios to explore

Arc Length — Chords shrink, √(1+f′²) emerges — measure a curve.

Real-world applications

Cable/chain lengths (catenary — bridges, power lines).
Road & rollercoaster track lengths from design curves.
Physics: distance along a trajectory from v(t) — same integral in parametric form.

JEE exam tips

cosh's arc length from 0 to a is sinh(a) — the cleanest exam case.
Parametric circles: √(ẋ² + ẏ²) = r ⇒ L = rθ, recovering s = rθ.
Distance travelled = ∫|v|dt — arc length of the position curve in time.

Common mistakes

Forgetting the 1 under the root (writing ∫y′dx).
Squaring y′ but rooting only part of the expression.
Using the x-form when the curve fails the vertical-line test (switch to parametric).

Exam traps to avoid

Arc length ≥ straight-line distance, always — sanity check.
y = x^{3/2} from 0 to 4: L = (8/27)(10^{3/2} − 1) — the classic doable integral.