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Arc length in polar coordinates practice problems

Arc length of a polar curve, L = ∫ √(r² + (dr/dθ)²) dθ. 28 problems with worked solutions; in 28 of them every equation is proved by a computer algebra system.

9 worked problems

Try each one before opening the solution. Every line is labeled with what checked it. How we verify

Problem 8.280

medium
Find the length of the polar curve \( \displaystyle r = 5 \cos{\left(\theta \right)} + 5 \) for \( \displaystyle 0 \le \theta \le \pi \).
  1. L = ∫ √(r² + (dr/dθ)²) dθ.
    Reviewed
  2. \[ \left(5 \cos{\left(\theta \right)} + 5\right)^{2} + \left(\frac{d}{d \theta} \left(5 \cos{\left(\theta \right)} + 5\right)\right)^{2} = 50 \cos{\left(\theta \right)} + 50 \]
    r² + (dr/dθ)², simplified.✓ Proved
  3. \[ 100 \cos^{2}{\left(\frac{\theta}{2} \right)} = 50 \cos{\left(\theta \right)} + 50 \]
    Its square root is 2a cos(θ/2) on [0, π] (half-angle identity).✓ Proved
  4. \[ \int\limits_{0}^{\pi} 10 \cos{\left(\frac{\theta}{2} \right)}\, d\theta = 20 \]
    Integrate.✓ Proved
Answer \( 20 \)

✓ Nihil obstat Lines: 3 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 8.282

medium
Find the length of the polar curve \( \displaystyle r = \theta^{2} \) for \( \displaystyle 0 \le \theta \le \sqrt{5} \).
  1. L = ∫ √(r² + (dr/dθ)²) dθ.
    Reviewed
  2. \[ \theta^{4} + \left(\frac{d}{d \theta} \theta^{2}\right)^{2} = \theta^{2} \left(\theta^{2} + 4\right) \]
    r² + (dr/dθ)², simplified.✓ Proved
  3. \[ \int\limits_{0}^{\sqrt{5}} \sqrt{\theta^{2} + 4} \left|{\theta}\right|\, d\theta = \frac{19}{3} \]
    Integrate.✓ Proved
Answer \( \frac{19}{3} \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 8.283

medium
Find the length of the polar curve \( \displaystyle r = e^{\theta} \) for \( \displaystyle 0 \le \theta \le 1 \).
  1. L = ∫ √(r² + (dr/dθ)²) dθ.
    Reviewed
  2. \[ e^{2 \theta} + \left(\frac{d}{d \theta} e^{\theta}\right)^{2} = 2 e^{2 \theta} \]
    r² + (dr/dθ)², simplified.✓ Proved
  3. \[ \int\limits_{0}^{1} \sqrt{2} e^{\theta}\, d\theta = \sqrt{2} \left(-1 + e\right) \]
    Integrate.✓ Proved
Answer \( \sqrt{2} \left(-1 + e\right) \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 8.284

medium
Find the length of the polar curve \( \displaystyle r = 3 \) for \( \displaystyle 0 \le \theta \le \frac{\pi}{2} \).
  1. L = ∫ √(r² + (dr/dθ)²) dθ.
    Reviewed
  2. \[ \left(\frac{d}{d \theta} 3\right)^{2} + 9 = 9 \]
    r² + (dr/dθ)², simplified.✓ Proved
  3. \[ \int\limits_{0}^{\frac{\pi}{2}} 3\, d\theta = \frac{3 \pi}{2} \]
    Integrate.✓ Proved
Answer \( \frac{3 \pi}{2} \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 8.285

medium
Find the length of the polar curve \( \displaystyle r = 5 \sin{\left(\theta \right)} \) for \( \displaystyle 0 \le \theta \le \pi \).
  1. L = ∫ √(r² + (dr/dθ)²) dθ.
    Reviewed
  2. \[ 25 \sin^{2}{\left(\theta \right)} + \left(\frac{d}{d \theta} 5 \sin{\left(\theta \right)}\right)^{2} = 25 \]
    r² + (dr/dθ)², simplified.✓ Proved
  3. \[ \int\limits_{0}^{\pi} 5\, d\theta = 5 \pi \]
    Integrate.✓ Proved
Answer \( 5 \pi \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 8.288

medium
Find the length of the polar curve \( \displaystyle r = \sin{\left(\theta \right)} \) for \( \displaystyle 0 \le \theta \le \pi \).
  1. L = ∫ √(r² + (dr/dθ)²) dθ.
    Reviewed
  2. \[ \sin^{2}{\left(\theta \right)} + \left(\frac{d}{d \theta} \sin{\left(\theta \right)}\right)^{2} = 1 \]
    r² + (dr/dθ)², simplified.✓ Proved
  3. \[ \int\limits_{0}^{\pi} 1\, d\theta = \pi \]
    Integrate.✓ Proved
Answer \( \pi \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 8.292

medium
Find the length of the polar curve \( \displaystyle r = \theta^{2} \) for \( \displaystyle 0 \le \theta \le 2 \).
  1. L = ∫ √(r² + (dr/dθ)²) dθ.
    Reviewed
  2. \[ \theta^{4} + \left(\frac{d}{d \theta} \theta^{2}\right)^{2} = \theta^{2} \left(\theta^{2} + 4\right) \]
    r² + (dr/dθ)², simplified.✓ Proved
  3. \[ \int\limits_{0}^{2} \sqrt{\theta^{2} + 4} \left|{\theta}\right|\, d\theta = - \frac{8}{3} + \frac{16 \sqrt{2}}{3} \]
    Integrate.✓ Proved
Answer \( - \frac{8}{3} + \frac{16 \sqrt{2}}{3} \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 8.406

medium
Find the length of the polar curve \( \displaystyle r = \cos{\left(\theta \right)} \) for \( \displaystyle 0 \le \theta \le \frac{\pi}{2} \).
  1. L = ∫ √(r² + (dr/dθ)²) dθ.
    Reviewed
  2. \[ \cos^{2}{\left(\theta \right)} + \left(\frac{d}{d \theta} \cos{\left(\theta \right)}\right)^{2} = 1 \]
    r² + (dr/dθ)², simplified.✓ Proved
  3. \[ \int\limits_{0}^{\frac{\pi}{2}} 1\, d\theta = \frac{\pi}{2} \]
    Integrate.✓ Proved
Answer \( \frac{\pi}{2} \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 8.409

medium
Find the length of the polar curve \( \displaystyle r = 2 \cos{\left(\theta \right)} \) for \( \displaystyle 0 \le \theta \le \frac{\pi}{2} \).
  1. L = ∫ √(r² + (dr/dθ)²) dθ.
    Reviewed
  2. \[ 4 \cos^{2}{\left(\theta \right)} + \left(\frac{d}{d \theta} 2 \cos{\left(\theta \right)}\right)^{2} = 4 \]
    r² + (dr/dθ)², simplified.✓ Proved
  3. \[ \int\limits_{0}^{\frac{\pi}{2}} 2\, d\theta = \pi \]
    Integrate.✓ Proved
Answer \( \pi \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

More arc length in polar coordinates practice

Find the length of the polar curve \( \displaystyle r = 6 \cos{\left(\theta \right)} + 6 \) for \( \displaystyle 0 \le \theta \le \pi \).
Problem 8.281medium✓ Nihil obstat
Find the length of the polar curve \( \displaystyle r = 1 \) for \( \displaystyle 0 \le \theta \le \frac{\pi}{2} \).
Problem 8.286medium✓ Nihil obstat
Find the length of the polar curve \( \displaystyle r = 2 \cos{\left(\theta \right)} + 2 \) for \( \displaystyle 0 \le \theta \le \pi \).
Problem 8.287medium✓ Nihil obstat
Find the length of the polar curve \( \displaystyle r = 4 \cos{\left(\theta \right)} + 4 \) for \( \displaystyle 0 \le \theta \le \pi \).
Problem 8.289medium✓ Nihil obstat
Find the length of the polar curve \( \displaystyle r = 2 \sin{\left(\theta \right)} \) for \( \displaystyle 0 \le \theta \le \pi \).
Problem 8.290medium✓ Nihil obstat
Find the length of the polar curve \( \displaystyle r = 6 \sin{\left(\theta \right)} \) for \( \displaystyle 0 \le \theta \le \pi \).
Problem 8.291medium✓ Nihil obstat
Find the length of the polar curve \( \displaystyle r = 5 \) for \( \displaystyle 0 \le \theta \le \frac{\pi}{2} \).
Problem 8.293medium✓ Nihil obstat
Find the length of the polar curve \( \displaystyle r = 3 \cos{\left(\theta \right)} + 3 \) for \( \displaystyle 0 \le \theta \le \pi \).
Problem 8.294medium✓ Nihil obstat
Find the length of the polar curve \( \displaystyle r = \theta^{2} \) for \( \displaystyle 0 \le \theta \le 1 \).
Problem 8.404medium✓ Every equation proved
Find the length of the polar curve \( \displaystyle r = 4 \sin{\left(\theta \right)} \) for \( \displaystyle 0 \le \theta \le \pi \).
Problem 8.405medium✓ Nihil obstat
Find the length of the polar curve \( \displaystyle r = 6 \) for \( \displaystyle 0 \le \theta \le \frac{\pi}{2} \).
Problem 8.407medium✓ Nihil obstat
Find the length of the polar curve \( \displaystyle r = 3 \sin{\left(\theta \right)} \) for \( \displaystyle 0 \le \theta \le \pi \).
Problem 8.408medium✓ Nihil obstat
Find the length of the polar curve \( \displaystyle r = 2 \) for \( \displaystyle 0 \le \theta \le \frac{\pi}{2} \).
Problem 8.410medium✓ Nihil obstat
Find the length of the polar curve \( \displaystyle r = 3 \cos{\left(\theta \right)} \) for \( \displaystyle 0 \le \theta \le \frac{\pi}{2} \).
Problem 8.411medium✓ Nihil obstat
Find the length of the polar curve \( \displaystyle r = 6 \cos{\left(\theta \right)} \) for \( \displaystyle 0 \le \theta \le \frac{\pi}{2} \).
Problem 8.412medium✓ Nihil obstat
Find the length of the polar curve \( \displaystyle r = 4 \) for \( \displaystyle 0 \le \theta \le \frac{\pi}{2} \).
Problem 8.413medium✓ Nihil obstat
Find the length of the polar curve \( \displaystyle r = \cos{\left(\theta \right)} + 1 \) for \( \displaystyle 0 \le \theta \le \pi \).
Problem 8.414medium✓ Every equation proved
Find the length of the polar curve \( \displaystyle r = 5 \cos{\left(\theta \right)} \) for \( \displaystyle 0 \le \theta \le \frac{\pi}{2} \).
Problem 8.415medium✓ Nihil obstat
Find the length of the polar curve \( \displaystyle r = 4 \cos{\left(\theta \right)} \) for \( \displaystyle 0 \le \theta \le \frac{\pi}{2} \).
Problem 8.416medium✓ Nihil obstat