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

검증된 단계별 안내

Recall the formula for the arc length of a parametric curve given by \(x = x(t)\) and \(y = y(t)\) over the interval \(a \leq t \leq b\): \[L = \int_a^b \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt\]

Identify the given functions: \(x(t) = \sin t\) \(y(t) = t - \cos t\) and the interval: \(0 \leq t \leq \frac{\pi}{2}\)

Compute the derivatives of \(x(t)\) and \(y(t)\) with respect to \(t\): \[\frac{dx}{dt} = \cos t\] \[\frac{dy}{dt} = 1 + \sin t\]

Substitute the derivatives into the arc length formula under the square root: \[\sqrt{(\cos t)^2 + (1 + \sin t)^2} = \sqrt{\cos^2 t + (1 + \sin t)^2}\]

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.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

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