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

x = eᵗ sin t, y = eᵗ cos t; 0 ≤ t ≤ 2π

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 \(\frac{dx}{dt}\) and \(\frac{dy}{dt}\) for the given functions: \[x = e^t \sin t, \quad y = e^t \cos t\] Use the product rule for differentiation since both \(x\) and \(y\) are products of \(e^t\) and trigonometric functions.

Compute \(\frac{dx}{dt}\): \[\frac{dx}{dt} = \frac{d}{dt}(e^t \sin t) = e^t \sin t + e^t \cos t\] Compute \(\frac{dy}{dt}\): \[\frac{dy}{dt} = \frac{d}{dt}(e^t \cos t) = e^t \cos t - e^t \sin t\]

Substitute \(\frac{dx}{dt}\) and \(\frac{dy}{dt}\) into the arc length formula under the square root: \[\sqrt{(e^t \sin t + e^t \cos t)^2 + (e^t \cos t - e^t \sin t)^2}\] Simplify the expression inside the square root by expanding and combining like terms.

Set up the integral for the arc length over the interval \(0 \leq t \leq 2\pi\): \[L = \int_0^{2\pi} \sqrt{(\frac{dx}{dt})^2 + (\frac{dy}{dt})^2} \, dt\] After simplification, evaluate or express the integral in a form ready for calculation.

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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 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 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 over the interval.

Differentiation of Exponential and Trigonometric Functions

Calculating dx/dt and dy/dt requires differentiating functions involving products of exponentials and trigonometric terms. Mastery of the product rule and derivatives of e^t, sin t, and cos t is necessary to correctly find these derivatives and proceed with the arc length calculation.

81–88. Arc length Find the arc length of the following curves on the given interval.x = eᵗ sin t, y = eᵗ cos t; 0 ≤ t ≤ 2π