Lengths of Curves

Find the lengths of the curves in Exercises 25–30.

x = cos t, y = t + sin t, 0 ≤ t ≤ π

검증된 단계별 안내

Recall the formula for the length of a curve defined parametrically 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 parametric functions: \[x(t) = \cos t, \quad y(t) = t + \sin t\] and the interval: \[0 \leq t \leq \pi\]

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

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

Simplify the expression inside the square root as much as possible to prepare for integration, then set up the integral for the length: \[L = \int_0^{\pi} \sqrt{\sin^2 t + (1 + \cos t)^2} \, dt\] From here, you can proceed to simplify and evaluate the integral to find the length.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

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 as t. In this problem, 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 length of a curve defined parametrically by x(t) and y(t) from t = a to t = b is found using the integral of the square root of the sum of the squares of the derivatives: L = ∫_a^b √[(dx/dt)² + (dy/dt)²] dt. This formula calculates the total distance traveled along the curve.

Differentiation of Trigonometric Functions

To apply the arc length formula, derivatives of x(t) and y(t) must be computed. Understanding how to differentiate trigonometric functions like sine and cosine is essential, as these derivatives determine the integrand for the arc length calculation.

Lengths of CurvesFind the lengths of the curves in Exercises 25–30.x = cos t, y = t + sin t, 0 ≤ t ≤ π