Question

The arc length of the one arch of the cycloid given by \( x = t - \sin t \) and \( y = 1 - \cos t \) is ………..

Hint:

1. For parametric curves, use the formula for arc length \( L = \int_a^b \sqrt{\left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2} \, dt \).
2. Simplify trigonometric expressions using identities like \( 1 - \cos t = 2\sin^2(t/2) \) to ease integration.

Answer 1

Step 1: Formula for arc length. The arc length of a parametric curve \( x(t) \) and \( y(t) \) from \( t = a \) to \( t = b \) is given by: \[ L = \int_a^b \sqrt{\left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2} \, dt \] Step 2: Derivatives of \( x \) and \( y \). For \( x = t - \sin t \) and \( y = 1 - \cos t \): \[ \frac{dx}{dt} = 1 - \cos t, \quad \frac{dy}{dt} = \sin t \] Step 3: Substitute into the formula. The square of the derivatives is: \[ \left( \frac{dx}{dt} \right)^2 = (1 - \cos t)^2, \quad \left( \frac{dy}{dt} \right)^2 = (\sin t)^2 \] Adding these: \[ \left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2 = (1 - \cos t)^2 + (\sin t)^2 \] Expand \( (1 - \cos t)^2 \): \[ (1 - \cos t)^2 = 1 - 2\cos t + \cos^2 t \] So: \[ \left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2 = 1 - 2\cos t + \cos^2 t + \sin^2 t \] Using \( \sin^2 t + \cos^2 t = 1 \), this simplifies to: \[ \left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2 = 2(1 - \cos t) \] Step 4: Simplify the arc length formula. The arc length becomes: \[ L = \int_0^{2\pi} \sqrt{2(1 - \cos t)} \, dt \] Using the trigonometric identity \( 1 - \cos t = 2\sin^2(t/2) \): \[ L = \int_0^{2\pi} \sqrt{2 \cdot 2\sin^2(t/2)} \, dt = \int_0^{2\pi} 2|\sin(t/2)| \, dt \] Step 5: Evaluate the integral. Since \( \sin(t/2) \) is non-negative in \( [0, 2\pi] \), we drop the absolute value: \[ L = \int_0^{2\pi} 2\sin(t/2) \, dt \] Let \( u = t/2 \), so \( du = dt/2 \) and the limits change to \( u = 0 \) to \( u = \pi \). The integral becomes: \[ L = 4 \int_0^\pi \sin u \, du \] The integral of \( \sin u \) is \( -\cos u \): \[ L = 4[-\cos u]_0^\pi = 4[-\cos \pi + \cos 0] = 4[-(-1) + 1] = 4(2) = 8 \] Conclusion: The arc length of one arch of the cycloid is \( \mathbf{8} \).