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.

Questions & Answers

Q: What is the formula for calculating the arc length of a polar curve?

The formula for calculating the arc length of a polar curve is the integral from α to β of the square root of (r^2 + (dr/dθ)^2) dθ, where r is the polar function and dr/dθ is its derivative.

Q: How do you find the arc length of a polar curve between given values of θ?

To find the arc length of a polar curve between given values of θ, substitute the limits of θ, the function r, and its derivative dr/dθ into the arc length formula. Then, integrate the expression and evaluate the integral.

Q: Can you explain how to find the arc length of a circle using the arc length formula?

Yes, to find the arc length of a circle using the arc length formula, substitute the radius r of the circle into the formula, which becomes 2πr for a full circle. For a half-circle, the arc length is πr.

Q: What is the arc length of a curve described by r = 2θ between θ = 0 and θ = π/2?

The arc length of the curve r = 2θ between θ = 0 and θ = π/2 is approximately 4.158 units, as calculated using the arc length formula and integration.

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.