Arc length:
Find the length of the curve y = x², 0 ≤ x ≤ √3/2.

Verified step by step guidance

1. Recall the formula for the arc length of a curve defined by y = f(x) from x = a to x = b: \[L = \int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx\]
2. Identify the function and the interval: here, \[y = x^2\] with \[a = 0\] and \[b = \frac{\sqrt{3}}{2}\]
3. Compute the derivative of y with respect to x: \[\frac{dy}{dx} = 2x\]
4. Substitute the derivative into the arc length formula: \[L = \int_{0}^{\frac{\sqrt{3}}{2}} \sqrt{1 + (2x)^2} \, dx = \int_{0}^{\frac{\sqrt{3}}{2}} \sqrt{1 + 4x^2} \, dx\]
5. Set up the integral for evaluation. To solve it, consider using a trigonometric substitution such as \[x = \frac{1}{2} \tan(\theta)\] or another suitable method to simplify the integral before integrating.

Key Concepts

Arc Length Formula for a Curve

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 infinitesimal line segments along the curve, accounting for both horizontal and vertical changes.

Derivative of the Function

To apply the arc length formula, you need the derivative dy/dx of the function y = x², which is 2x. The derivative represents the slope of the tangent line at any point and is essential for calculating the integrand √(1 + (dy/dx)²).

Definite Integration

After substituting the derivative into the arc length formula, you evaluate the definite integral from the lower limit 0 to the upper limit √3/2. This process computes the exact length of the curve segment between these points.