How do you find the exact length of the polar curve defined by the equation r = 0.8e^(2θ)?

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To find the exact length of a polar curve defined by the equation r = 0.8e^{2θ}, we need to use the formula for the length of a polar curve:

L = ∫ (from α to β) √(r^2 + (dr/dθ)^2) dθ

Here, r is the polar coordinate function, dr/dθ is the derivative of r with respect to θ, and α and β are the angles that bound the portion of the curve we want to measure.

Steps to Calculate the Length:

  1. First, find dr/dθ:
    • Given r = 0.8e^{2θ}, differentiate it with respect to θ:
      • dr/dθ = 0.8 * 2e^{2θ} = 1.6e^{2θ}
  2. Next, calculate √(r^2 + (dr/dθ)^2):
    • Substituting r and dr/dθ, we get:
      • r^2 = (0.8e^{2θ})^2 = 0.64e^{4θ}
      • (dr/dθ)^2 = (1.6e^{2θ})^2 = 2.56e^{4θ}
    • Thus,
      • √(r^2 + (dr/dθ)^2) = √(0.64e^{4θ} + 2.56e^{4θ}) = √(3.2e^{4θ}) = e^{2θ}√3.2
  3. Set the limits of integration α and β:
    • These values depend on the part of the curve you want to measure. For example, if you want to measure one full cycle, you could use α = 0 and β = π.
  4. Integrate:
    • Plugging everything into the integral, we get:
      • L = ∫ (from 0 to π) e^{2θ}√3.2 dθ
    • This simplifies to:
      • L = √3.2 ∫ (from 0 to π) e^{2θ} dθ
    • Calculating this integral yields:
      • ∫ e^{2θ} dθ = (1/2)e^{2θ}
      • Evaluating from 0 to π gives:
    • Bringing it all together, the length of the polar curve from θ = 0 to θ = π becomes:
      • L = (√3.2)/2 (e^{2π} – 1)

In summary, the exact length of the polar curve defined by r = 0.8e^{2θ} over the interval [0, π] is given by:

L = (√3.2)/2 (e^{2π} – 1)

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