81–88. Arc length Find the arc length of the following curves on the given interval.


x = sin t, y = t - cos t; 0 ≤ t ≤ π/2

Verified step by step guidance

1. Recall the formula for the arc length of a parametric curve given by \(x = x(t)\) and \(y = y(t)\) over the interval \(a \leq t \leq b\): \[L = \int_a^b \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt\]
2. Identify the given functions: \(x(t) = \sin t\) \(y(t) = t - \cos t\) and the interval: \(0 \leq t \leq \frac{\pi}{2}\)
3. Compute the derivatives of \(x(t)\) and \(y(t)\) with respect to \(t\): \[\frac{dx}{dt} = \cos t\] \[\frac{dy}{dt} = 1 + \sin t\]
4. Substitute the derivatives into the arc length formula under the square root: \[\sqrt{(\cos t)^2 + (1 + \sin t)^2} = \sqrt{\cos^2 t + (1 + \sin t)^2}\]
5. Simplify the expression inside the square root as much as possible before integrating, then set up the integral for the arc length: \[L = \int_0^{\frac{\pi}{2}} \sqrt{\cos^2 t + (1 + \sin t)^2} \, dt\] This integral can then be evaluated to find the arc length.

Key Concepts

Parametric Equations

Parametric equations express the coordinates of points on a curve as functions of a parameter, often denoted t. Here, x and y are given in terms of t, allowing the curve to be analyzed by studying these functions over the specified interval.

Arc Length Formula for Parametric Curves

The arc length of a curve defined parametrically by x(t) and y(t) from t = a to t = b is found by integrating the square root of the sum of the squares of the derivatives: ∫ from a to b √[(dx/dt)² + (dy/dt)²] dt. This formula measures the distance along the curve.

Differentiation of Parametric Functions

To apply the arc length formula, one must compute the derivatives dx/dt and dy/dt accurately. Differentiation rules for trigonometric and polynomial functions are used here to find these derivatives, which are essential for evaluating the integral.