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


b. If necessary, use technology to evaluate or approximate the integral.
y = cos 2x, for 0 ≤ x ≤ π

Verified step by step guidance

1. Recall the formula for the arc length of a curve defined by a function \( y = f(x) \) from \( x = a \) to \( x = b \): \[ L = \int_a^b \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx \]
2. Identify the function and the interval: here, \( y = \cos(2x) \) and the interval is \( 0 \leq x \leq \pi \).
3. Compute the derivative \( \frac{dy}{dx} \) of \( y = \cos(2x) \) using the chain rule: \[ \frac{dy}{dx} = -2 \sin(2x) \]
4. Substitute \( \frac{dy}{dx} \) into the arc length formula to get the integral: \[ L = \int_0^{\pi} \sqrt{1 + (-2 \sin(2x))^2} \, dx = \int_0^{\pi} \sqrt{1 + 4 \sin^2(2x)} \, dx \]
5. Since this integral is not straightforward to solve analytically, use a calculator or appropriate technology to approximate the value of the integral over the interval \( [0, \pi] \).

Key Concepts

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 distance along the curve by summing infinitesimal line segments, requiring the derivative of the function.

Derivative of the Function

To find the arc length, you must compute dy/dx, the derivative of y with respect to x. For y = cos(2x), use the chain rule: dy/dx = -2 sin(2x). This derivative is then squared and used inside the arc length integral.

Use of Technology for Integration

Some integrals, like the arc length integral for y = cos(2x), may not have simple antiderivatives. Technology such as graphing calculators or computer algebra systems can approximate or evaluate these integrals numerically, providing practical solutions.