Functions from arc length What differentiable functions have an arc length on the interval [a, b] given by the following integrals? Note that the answers are not unique. Give a family of functions that satisfy the conditions.
b. ∫a^b √1+36 cos² 2xdx

Verified step by step guidance

1. Recall the formula for the arc length of a differentiable function \( y = f(x) \) on the interval \([a,b]\): \[ L = \int_a^b \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx \]
2. Compare the given integral for arc length: \[ \int_a^b \sqrt{1 + 36 \cos^2(2x)} \, dx \] with the general formula. This means: \[ 1 + \left(\frac{dy}{dx}\right)^2 = 1 + 36 \cos^2(2x) \]
3. From the equality above, isolate \( \frac{dy}{dx} \): \[ \left(\frac{dy}{dx}\right)^2 = 36 \cos^2(2x) \] which implies \[ \frac{dy}{dx} = \pm 6 \cos(2x) \]
4. Integrate \( \frac{dy}{dx} = \pm 6 \cos(2x) \) with respect to \( x \) to find the family of functions: \[ y = \pm 6 \int \cos(2x) \, dx + C \]
5. Recall that \( \int \cos(2x) \, dx = \frac{1}{2} \sin(2x) + C \), so the family of functions is: \[ y = \pm 3 \sin(2x) + C \] where \( C \) is an arbitrary constant.

Key Concepts

Arc Length Formula

The arc length of a differentiable function y = f(x) on [a, b] is given by the integral ∫_a^b √(1 + (f'(x))²) dx. This formula measures the length of the curve by summing infinitesimal line segments, where the integrand involves the derivative of the function.

Relating the Integrand to the Derivative

To find functions with a given arc length integral, identify the expression inside the square root as 1 + (f'(x))². Equate this to the given integrand to solve for f'(x), which helps determine the family of functions whose derivatives satisfy the condition.

Solving Differential Equations for Families of Functions

Once f'(x) is found, integrate it to obtain the general form of f(x). Since integration introduces an arbitrary constant, the solution represents a family of functions, not a unique one, matching the problem's requirement for multiple solutions.