Length of polar curve formula is a mathematical expression used to calculate the distance along a polar curve, which is defined in the polar coordinate system. The formula involves four key entities: the radius vector, the angle of rotation, the derivative of the radius vector, and the derivative of the angle of rotation.
Length of Polar Curve Formula
The formula for the length of a polar curve is:
L = ∫[a,b] √(r² + (dr/dθ)²) dθ
where:
- L is the length of the curve
- a and b are the starting and ending angles of the curve
- r is the polar coordinate function
- dr/dθ is the derivative of r with respect to θ
This formula can be derived from the following steps:
- Imagine the polar curve as a series of small line segments.
- The length of each line segment is given by the following formula:
Δs = √(Δr² + r² Δθ²)
where:
- Δs is the length of the line segment
- Δr is the change in r
- r is the average value of r over the interval Δθ
- Δθ is the change in θ
- The total length of the curve is the sum of the lengths of all the line segments:
L = ∫[a,b] √(Δr² + r² Δθ²) dθ
- We can simplify this formula by using the following trigonometric identity:
sin² θ + cos² θ = 1
This identity allows us to write the following:
Δr² + r² Δθ² = r² (Δθ)² + Δr²
- We can now substitute this expression into the formula for the length of the curve:
L = ∫[a,b] √(r² (Δθ)² + Δr²) dθ
- We can now take the limit as Δθ approaches 0:
L = ∫[a,b] √(r² + (dr/dθ)²) dθ
Example: Finding the Length of a Polar Curve
Find the length of the polar curve r = 2cos θ from θ = 0 to θ = π/2.
L = ∫[0,π/2] √(2²cos² θ + (-2sin θ)²) dθ
= ∫[0,π/2] √(4cos² θ + 4sin² θ) dθ
= ∫[0,π/2] √4 dθ
= 4∫[0,π/2] dθ
= 4[θ]_{0}^{π/2}
= 4(π/2 - 0)
= 2π
Therefore, the length of the curve is 2π.
Question 1:
How to calculate the length of a polar curve?
Answer:
The length of a polar curve given by the equation r = f(θ) over the interval [α, β] is calculated using the formula:
$ L = \int_{\alpha}^{\beta} \sqrt{r^2 + (\frac{dr}{d\theta})^2} d\theta $
Question 2:
Why is the integral used in the length of polar curve formula?
Answer:
The integral is used to compute the length of the curve because it represents the limit of the sum of the lengths of small line segments that approximate the curve.
Question 3:
What is the significance of the derivative in the length of polar curve formula?
Answer:
The derivative of r with respect to θ, dr/dθ, represents the slope of the tangent line to the curve at any given point. It is squared and added to r^2 to account for the vertical and horizontal components of the curve’s length.
Thanks for sticking with me for this math adventure! I know it can be a bit mind-boggling at times, but hopefully, you’ve gained a newfound appreciation for the beauty and complexity of polar curves. If you’re ever feeling curious or need a refresher, feel free to swing by again. I’ll always be here to help you unravel the mysteries of calculus. Until next time, keep your pencils sharp and your minds open!