58–61. Arc length Find the length of the following curves.
y = x³/6 + 1/2x on [1,2]

Verified step by step guidance

1. Identify the function given: \(y = \frac{x^3}{6} + \frac{1}{2}x\) and the interval \([1, 2]\) over which we want to find the arc length.
2. Recall the formula for the arc length \(L\) of a curve \(y = f(x)\) from \(x = a\) to \(x = b\): \(L = \int_a^b \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx\).
3. Compute the derivative \(\frac{dy}{dx}\) of the function: \(\frac{dy}{dx} = \frac{d}{dx} \left( \frac{x^3}{6} + \frac{1}{2}x \right)\).
4. Square the derivative to get \(\left(\frac{dy}{dx}\right)^2\) and then add 1 inside the square root: \(1 + \left(\frac{dy}{dx}\right)^2\).
5. Set up the integral for the arc length: \(L = \int_1^2 \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx\) and prepare to evaluate or simplify this integral.

Key Concepts

Arc Length Formula

The arc length of a curve y = f(x) from x = a to x = b is found using the integral formula L = ∫_a^b √(1 + (dy/dx)²) dx. This formula sums the lengths of infinitesimal line segments along the curve, providing the total distance traveled along it.

Derivative of the Function

To apply the arc length formula, you need the derivative dy/dx of the function y = x³/6 + 1/2 x. The derivative represents the slope of the curve at any point and is essential for calculating the integrand √(1 + (dy/dx)²).

Definite Integration

After finding the integrand, you evaluate the definite integral from x = 1 to x = 2. This process involves integrating the function √(1 + (dy/dx)²) over the interval to compute the exact length of the curve segment.