21–30. {Use of Tech} Arc length by calculator

a. Write and simplify the integral that gives the arc length of the following curves on the given interval.
y = cos 2x, for 0 ≤ x ≤ π

검증된 단계별 안내

Recall the formula for the arc length of a curve defined by 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 and interval: here, \[y = \cos 2x\] with \[0 \leq x \leq \pi\]

Find the derivative of y with respect to x: \[\frac{dy}{dx} = \frac{d}{dx}(\cos 2x) = -2 \sin 2x\]

Square the derivative: \[\left(\frac{dy}{dx}\right)^2 = (-2 \sin 2x)^2 = 4 \sin^2 2x\]

Write the integral for the arc length using the formula: \[L = \int_{0}^{\pi} \sqrt{1 + 4 \sin^2 2x} \, dx\] This integral represents the arc length of the curve on the given interval.

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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 ∫ from a to b of √(1 + (dy/dx)²) dx. This formula calculates the length of the curve by summing infinitesimal line segments along the curve.

Derivative of the Function

To apply the arc length formula, you need the derivative dy/dx of the function y = cos(2x). Differentiating y with respect to x gives dy/dx = -2 sin(2x), which is essential for substituting into the arc length integral.

Simplifying the Integral Expression

After finding dy/dx, substitute it into the arc length integral and simplify the expression under the square root. Simplification may involve trigonometric identities to make the integral easier to evaluate, especially when using a calculator.

21–30. {Use of Tech} Arc length by calculatora. Write and simplify the integral that gives the arc length of the following curves on the given interval. y = cos 2x, for 0 ≤ x ≤ π