Lengths of Curves

Find the lengths of the curves in Exercises 13–19.

x = 5 cos t − cos 5t, y = 5 sin t − sin 5t, 0 ≤ t ≤ π/2

검증된 단계별 안내

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

Identify the given functions: \[x(t) = 5 \cos t - \cos 5t\] \[y(t) = 5 \sin t - \sin 5t\] and the interval is \(0 \leq t \leq \frac{\pi}{2}\).

Compute the derivatives \(\frac{dx}{dt}\) and \(\frac{dy}{dt}\): For \(x(t)\), use the derivative of cosine: \[\frac{dx}{dt} = -5 \sin t + 5 \sin 5t\] For \(y(t)\), use the derivative of sine: \[\frac{dy}{dt} = 5 \cos t - 5 \cos 5t\]

Substitute these derivatives into the arc length formula under the square root: \[L = \int_0^{\frac{\pi}{2}} \sqrt{\left(-5 \sin t + 5 \sin 5t\right)^2 + \left(5 \cos t - 5 \cos 5t\right)^2} \, dt\]

Simplify the expression inside the square root if possible, then set up the integral for evaluation. This integral may require numerical methods or further trigonometric simplifications to solve.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

14m

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. Instead of y as a function of x, both x and y depend on t, allowing the description of more complex curves. Understanding how to work with parametric forms is essential for calculating properties like curve length.

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 by integrating the square root of the sum of the squares of the derivatives: ∫ₐᵇ √[(dx/dt)² + (dy/dt)²] dt. This formula generalizes the Pythagorean theorem to infinitesimal segments along the curve.

Differentiation of Trigonometric Functions

Calculating the derivatives dx/dt and dy/dt requires knowledge of differentiating trigonometric functions like sine and cosine. Recognizing the derivatives (d/dt) sin t = cos t and (d/dt) cos t = -sin t is crucial for correctly applying the arc length formula to the given parametric equations.

Lengths of CurvesFind the lengths of the curves in Exercises 13–19.x = 5 cos t − cos 5t, y = 5 sin t − sin 5t, 0 ≤ t ≤ π/2