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

Guida verificata passo dopo passo

Recall the formula for the arc length of a curve defined by y = f(x) from x = a to x = b: \[L = \int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx\]

Identify the function and the interval: here, \[y = x^2\] with \[a = 0\] and \[b = \frac{\sqrt{3}}{2}\]

Compute the derivative of y with respect to x: \[\frac{dy}{dx} = 2x\]

Substitute the derivative into the arc length formula: \[L = \int_{0}^{\frac{\sqrt{3}}{2}} \sqrt{1 + (2x)^2} \, dx = \int_{0}^{\frac{\sqrt{3}}{2}} \sqrt{1 + 4x^2} \, dx\]

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.

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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.

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Arc Length of Parametric Curves

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).

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Derivatives of Other Trig Functions

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.

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Percorso guidato