In Exercises 139–142, find the length of each curve.
141. y = ln(cos(x)) from x = 0 to x = π/4.

검증된 단계별 안내

Recall the formula for the length of a curve defined by a function \(y = f(x)\) from \(x = a\) to \(x = b\): \[L = \int_a^b \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx\]

Identify the function given: \[y = \ln(\cos(x))\] with the interval from \(x = 0\) to \(x = \frac{\pi}{4}\).

Find the derivative \(\frac{dy}{dx}\) using the chain rule: Since \(y = \ln(u)\) where \(u = \cos(x)\), then \[\frac{dy}{dx} = \frac{1}{u} \cdot \frac{du}{dx} = \frac{1}{\cos(x)} \cdot (-\sin(x)) = -\tan(x)\]

Substitute \(\frac{dy}{dx} = -\tan(x)\) into the arc length formula: \[L = \int_0^{\frac{\pi}{4}} \sqrt{1 + (-\tan(x))^2} \, dx = \int_0^{\frac{\pi}{4}} \sqrt{1 + \tan^2(x)} \, dx\]

Use the trigonometric identity \(1 + \tan^2(x) = \sec^2(x)\) to simplify the integrand: \[L = \int_0^{\frac{\pi}{4}} \sqrt{\sec^2(x)} \, dx = \int_0^{\frac{\pi}{4}} |\sec(x)| \, dx\] Since \(\sec(x)\) is positive on \([0, \frac{\pi}{4}]\), this becomes \[L = \int_0^{\frac{\pi}{4}} \sec(x) \, dx\] The next step would be to evaluate this integral.

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.

Arc Length Formula

The arc length of a curve y = f(x) from x = a to x = b is given by the integral L = ∫_a^b √(1 + (dy/dx)^2) dx. This formula calculates the length by summing infinitesimal line segments along the curve.

Derivative of y = ln(cos(x))

To find the arc length, you need the derivative dy/dx. For y = ln(cos(x)), use the chain rule: dy/dx = -tan(x), since the derivative of ln(u) is 1/u * du/dx and d/dx[cos(x)] = -sin(x).

Integration Techniques for Arc Length

Evaluating the arc length integral often requires simplifying the integrand and applying appropriate integration methods, such as substitution or recognizing standard integral forms, to compute the exact length over the given interval.

In Exercises 139–142, find the length of each curve.141. y = ln(cos(x)) from x = 0 to x = π/4.