22–23. Arc length Find the length of the following curves.
x = cos 2t, y = 2t - sin 2t; 0 ≤ t ≤ π/4

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 \(a \leq t \leq b\): \[L = \int_a^b \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt\]

Find the derivatives of the parametric functions with respect to \(t\): Calculate \(\frac{dx}{dt}\) for \(x = \cos 2t\) and \(\frac{dy}{dt}\) for \(y = 2t - \sin 2t\).

Substitute the derivatives into the arc length formula under the square root: \[\sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2}\]

Simplify the expression inside the square root as much as possible to make the integral easier to evaluate.

Set up the definite integral for the arc length from \(t = 0\) to \(t = \frac{\pi}{4}\): \[L = \int_0^{\frac{\pi}{4}} \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt\] Then, evaluate the integral to find the length of the curve.

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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 as 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 these is essential for calculating properties like arc length.

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 given by the integral of the square root of (dx/dt)² + (dy/dt)² dt. This formula sums the infinitesimal distances along the curve, providing the total length between the parameter limits.

Differentiation of Trigonometric Functions

Calculating dx/dt and dy/dt requires differentiating trigonometric functions like cosine and sine. Knowing the derivatives of these functions and applying the chain rule correctly is crucial to find the velocity components needed for the arc length integral.

22–23. Arc length Find the length of the following curves.x = cos 2t, y = 2t - sin 2t; 0 ≤ t ≤ π/4