How to Calculate the Arc Length of Polar Curves

TL;DR

To calculate the arc length of polar curves, use the formula: arc length = ∫√(r² + (dr/dθ)²) dθ. For example, for r = 6sinθ from 0 to π/2, the arc length is 3π units. Similarly, for r = 5 over 0 to 2π, the arc length is 10π units.

Transcript

so let's say that r is equal to six sine theta and our goal is to find the length of the polar curve over the interval where theta is between zero and pi over 2. so what's the formula that we need to find the arc length the arc length is going to be the integral from alpha to beta square root r squared plus d r d theta squared and then d theta so w... Read More

Key Insights

• 🫠 The arc length of a polar curve can be calculated using integration and the arc length formula, which involves the function r and its derivative.
• 😑 Arc length can be found between specific values of θ by substituting them into the formula and integrating the expression.
• ⭕ The arc length of a circle is equal to the circumference of the circle, which is 2π times the radius.

Summary & Key Takeaways

• The video demonstrates the process of finding the arc length of a polar curve by using the formula: square root of (r^2 + (dr/dθ)^2) dθ.

• It provides an example of finding the arc length of a curve defined by r = 6sinθ over the interval θ ∈ [0, π/2], resulting in an arc length of 3π units.

• The video also shows how to find the arc length of a circle with r = 5 over the interval θ ∈ [0, 2π], resulting in an arc length of 10π units.

• Lastly, it presents a spiral curve defined by r = 2θ and calculates its arc length over the interval θ ∈ [0, π/2] to be approximately 4.158 units.