What are the conditions required for the arc length formula to be valid?

Conditions

• The curve is defined by a function $y = f(x)$.
• The interval of integration is $[a, b]$.
• $f(x)$ is continuous on $[a, b]$.
• $f'(x)$ is continuous on $[a, b]$.

Reasoning, step by step

1. Identify the function $f(x)$ that defines the curve.
2. Verify that $f(x)$ is continuous over the specified interval $[a, b]$.
3. Calculate the derivative $f'(x)$.
4. Verify that $f'(x)$ is continuous over the same interval $[a, b]$.
5. Apply the arc length formula $\int_{a}^{b} \sqrt{1 + (f'(x))^2} dx$ only if both continuity conditions are met.

Example

The video explicitly lists the conditions for the Arc Length Formula knowledge item: "$f(x)$ must be continuous on [a, b]" and "f'(x) must be continuous on [a, b]". In the worked example, $f(x) = x^{3/2}$ and $f'(x) = \frac{3}{2}x^{1/2}$ are both continuous on $[0, 32/9]$.

Common misconceptions

• Assuming the formula works for any curve, including those with sharp corners or discontinuities.
• Forgetting to check the continuity of the derivative $f'(x)$, focusing only on $f(x)$.
• Applying the formula to parametric or polar curves without converting them to the appropriate $y=f(x)$ form or using their specific arc length formulas.

Watch the explanation

Explore next

• How do you set up the definite integral to find the arc length of the curve $y = x^{3/2}$ over the interval [0, $32/9$]?
• Why is the substitution $u = 1 + 9/4x$ chosen for this arc-length integral?