Arc length: Find the length of the curve y = ln(sec x), 0 ≤ x ≤ π/4.

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Recall the formula for the arc 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: \(y = \ln(\sec x)\). We need to find its derivative \(\frac{dy}{dx}\).

Compute the derivative \(\frac{dy}{dx}\) using the chain rule and the derivative of \(\sec x\): \[\frac{dy}{dx} = \frac{d}{dx} \ln(\sec x) = \frac{1}{\sec x} \cdot \sec x \tan x = \tan x\]

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

Simplify the integrand using the trigonometric identity \(1 + \tan^2 x = \sec^2 x\): \[L = \int_0^{\pi/4} \sqrt{\sec^2 x} \, dx = \int_0^{\pi/4} \sec x \, dx\] Now, set up the integral of \(\sec x\) from \(0\) to \(\pi/4\) to find the arc length.

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주요 개념

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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, requiring the derivative of the function.

Derivative of y = ln(sec x)

To apply the arc length formula, we need dy/dx. For y = ln(sec x), use the chain rule and the derivative of sec x, which is sec x tan x. The derivative simplifies to tan x, which is essential for evaluating the integral.

Integration Techniques for Arc Length

After substituting dy/dx into the arc length integral, the resulting integral often involves trigonometric identities. Simplifying the integrand using identities like 1 + tan^2 x = sec^2 x helps evaluate the integral more easily.