Compute the Arc Length of a Curve

Question

This question asks the candidate to compute the arc length of a given curve using the appropriate mathematical formulas.

Codemia · Accepted Answer

To compute the arc length of a curve, we need to understand the formula for arc length in a Cartesian coordinate system. The arc length $ L $ of a curve defined by a function $ y = f(x) $ from $ x = a $ to $ x = b $ is given by the formula:

$$ L = \int_{a}^{b} \sqrt{1 + (f'(x))^2} \, dx $$

Here, $ f'(x) $ is the derivative of the function.

First, we identify the curve's equation. For example, if the curve is given by $ y = f(x) $, we will need to compute its derivative $ f'(x) $. Next, we will substitute this derivative into the arc length formula. We also need to define the limits of integration, which are the x-values over which we want to compute the arc length.

Let’s say the curve is $ y = x^2 $ from $ x = 0 $ to $ x = 1 $. First, we compute the derivative:

$$ f'(x) = 2x $$

Next, we substitute this into the arc length formula:

$$ L = \int_{0}^{1} \sqrt{1 + (2x)^2} \, dx = \int_{0}^{1} \sqrt{1 + 4x^2} \, dx $$

Now, we evaluate this integral using appropriate techniques such as substitution or numerical methods if necessary.

When computing arc lengths for more complex curves, the integration may not be straightforward. In such cases, numerical integration methods (like Simpson's rule or trapezoidal rule) can be employed for approximation. Additionally, the choice of limits can significantly affect the result, so careful consideration is required to ensure accuracy.

After performing the integration, we arrive at the final arc length value. It is crucial to present this answer clearly, indicating the units of measurement, if applicable. For example, if the arc length is computed to be approximately 1.47 units, we would state:

"The arc length of the curve $ y = x^2 $ from $ x = 0 $ to $ x = 1 $ is approximately 1.47 units."

Compute the Arc Length of a Curve

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