Question

$$ \begin{array}{l}{\text { Multiple Choice Which of the following gives the best }} \\ {\text { approximation of the length of the arc of } y=\cos (2 x) \text { from } x=0} \\ {\text { to } x=\pi / 4 ? \quad }\end{array} $$ (A) 0.785 (B) 0.955 (C) 1.0 (D) 1.318 (E) 1.977

Short answer

C = 1.0 is the correct approximation for the length of the arc. As with any definite integral calculation, slight variations in the decimal value may occur due to the precision of the numerical approximation method used.

Step-by-step solution

Identify the Formula

The formula for the arc length of the curve \(y=f(x)\) from \(x=a\) to \(x=b\) is given by: \(\int_{a}^{b} \sqrt{1 + [f'(x)]^{2}} dx \) Where \(f'(x)\) denotes the derivative of the function.

Substitute in the given function

The given function is \(y= cos(2x)\). The derivative of this function is \(f'(x) = -2sin(2x)\). Substituting these into the arc length formula, we get : \(\int_{0}^{\pi/4} \sqrt{1 + [-2sin(2x)]^{2}} dx\).

Simplify and calculate the integral

Squaring and simplifying under the square root, we get: \(\int_{0}^{\pi/4} \sqrt{1 + 4sin^{2}(2x)} dx\). The value of the definite integral can then be approximated using numerical methods or by looking it up in mathematical tables.