Arc Length Calculator Calc 2

Polar Arc Length Formula:

\[ L = \int_{\theta_1}^{\theta_2} \sqrt{r^2 + \left(\frac{dr}{d\theta}\right)^2} d\theta \]

Arc Length (L):

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1. What is Polar Arc Length?

The arc length of a polar curve r(θ) between angles θ₁ and θ₂ measures the distance along the curve. It's calculated by integrating the square root of r² plus the derivative of r with respect to θ squared.

2. How Does the Calculator Work?

The calculator uses the polar arc length formula:

\[ L = \int_{\theta_1}^{\theta_2} \sqrt{r^2 + \left(\frac{dr}{d\theta}\right)^2} d\theta \]

Where:

\( L \) — Arc length (length units)
\( r(θ) \) — Polar function
\( \frac{dr}{dθ} \) — Derivative of r with respect to θ
\( θ \) — Angle in radians

Explanation: The formula accounts for both the radial distance and the angular change to calculate the true length along the curve.

3. Importance of Arc Length Calculation

Details: Calculating arc length is essential in physics, engineering, and computer graphics for determining distances along curved paths and parameterizing curves by arc length.

4. Using the Calculator

Tips: Enter the polar function r(θ), start angle θ₁, and end angle θ₂ in radians. The function should be differentiable between the given angles.

5. Frequently Asked Questions (FAQ)

Q1: What types of polar functions can I enter?
A: The calculator supports standard polar functions like circles, cardioids, spirals, and roses, provided they are differentiable in the interval.

Q2: Why does the formula include the derivative?
A: The derivative accounts for how quickly the radius changes with angle, which affects the actual path length.

Q3: Can I use degrees instead of radians?
A: The formula requires angles in radians. Convert degrees to radians by multiplying by π/180.

Q4: What if my function has vertical tangents?
A: The integral may become improper at points where dr/dθ is undefined. Special techniques may be needed for such cases.

Q5: How accurate is the numerical integration?
A: Accuracy depends on the integration method used and the complexity of the function.