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Arc length practice problems

The length of a curve: ∫ √(1 + (dy/dx)²) dx. 57 problems with worked solutions; in 57 of them every equation is proved by a computer algebra system.

3 worked problems

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

Problem 5.51

hard
Find the length of the curve \( \displaystyle y = \frac{2 \left(x + 3\right)^{\frac{3}{2}}}{3} \) from \( \displaystyle x = 1 \) to \( \displaystyle x = 2 \).
  1. Arc length is ∫ √(1 + (dy/dx)²) dx.
    Reviewed
  2. \[ \frac{d}{d x} \frac{2 \left(x + 3\right)^{\frac{3}{2}}}{3} = \sqrt{x + 3} \]
    dy/dx.✓ Proved
  3. \[ x + 4 \]
    1 + (dy/dx)² simplifies (here to a perfect square, which is why these are set).Reviewed
  4. \[ \int\limits_{1}^{2} \sqrt{x + 4}\, dx = - \frac{10 \sqrt{5}}{3} + 4 \sqrt{6} \]
    Integrate.✓ Proved
Answer \( - \frac{10 \sqrt{5}}{3} + 4 \sqrt{6} \)

✓ Nihil obstat Lines: 2 proved, 2 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 5.53

hard
Find the length of the curve \( \displaystyle y = \frac{2 x^{\frac{3}{2}}}{3} \) from \( \displaystyle x = 2 \) to \( \displaystyle x = 4 \).
  1. Arc length is ∫ √(1 + (dy/dx)²) dx.
    Reviewed
  2. \[ \frac{d}{d x} \frac{2 x^{\frac{3}{2}}}{3} = \sqrt{x} \]
    dy/dx.✓ Proved
  3. \[ x + 1 \]
    1 + (dy/dx)² simplifies (here to a perfect square, which is why these are set).Reviewed
  4. \[ \int\limits_{2}^{4} \sqrt{x + 1}\, dx = - 2 \sqrt{3} + \frac{10 \sqrt{5}}{3} \]
    Integrate.✓ Proved
Answer \( - 2 \sqrt{3} + \frac{10 \sqrt{5}}{3} \)

✓ Nihil obstat Lines: 2 proved, 2 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 5.49

medium
Find the length of the curve \( \displaystyle y = \frac{x^{4}}{8} + \frac{1}{4 x^{2}} \) from \( \displaystyle x = 1 \) to \( \displaystyle x = 3 \).
  1. Arc length is ∫ √(1 + (dy/dx)²) dx.
    Reviewed
  2. \[ \frac{d}{d x} \left(\frac{x^{4}}{8} + \frac{1}{4 x^{2}}\right) = \frac{x^{3}}{2} - \frac{1}{2 x^{3}} \]
    dy/dx.✓ Proved
  3. \[ \left(\frac{x^{3}}{2} - \frac{1}{2 x^{3}}\right)^{2} + 1 = \frac{\left(x^{2} + 1\right)^{2} \left(x^{4} - x^{2} + 1\right)^{2}}{4 x^{6}} \]
    1 + (dy/dx)² simplifies (here to a perfect square, which is why these are set).✓ Proved
  4. \[ \int\limits_{1}^{3} \sqrt{\left(\frac{x^{3}}{2} - \frac{1}{2 x^{3}}\right)^{2} + 1}\, dx = \frac{92}{9} \]
    Integrate.✓ Proved
Answer \( \frac{92}{9} \)

✓ 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

More arc length practice

Find the length of the curve \( \displaystyle y = \frac{2 \left(x + 1\right)^{\frac{3}{2}}}{3} \) from \( \displaystyle x = 2 \) to \( \displaystyle x = 7 \).
Problem 5.48medium✓ Nihil obstat
Find the length of the curve \( \displaystyle y = \frac{x^{4}}{8} + \frac{1}{4 x^{2}} \) from \( \displaystyle x = 2 \) to \( \displaystyle x = 3 \).
Problem 5.50medium✓ Nihil obstat
Find the length of the curve \( \displaystyle y = \frac{2 \left(x + 2\right)^{\frac{3}{2}}}{3} \) from \( \displaystyle x = 1 \) to \( \displaystyle x = 2 \).
Problem 5.54medium✓ Nihil obstat
Find the length of the curve \( \displaystyle y = \frac{2 \left(x + 1\right)^{\frac{3}{2}}}{3} \) from \( \displaystyle x = 1 \) to \( \displaystyle x = 4 \).
Problem 5.57medium✓ Nihil obstat
Find the length of the curve \( \displaystyle y = \frac{x^{4}}{8} + \frac{1}{4 x^{2}} \) from \( \displaystyle x = 1 \) to \( \displaystyle x = 2 \).
Problem 5.58medium✓ Nihil obstat
Find the length of the curve \( \displaystyle y = \frac{x^{4}}{8} + \frac{1}{4 x^{2}} \) from \( \displaystyle x = 1 \) to \( \displaystyle x = 4 \).
Problem 5.60medium✓ Every equation proved
Find the length of the curve \( \displaystyle y = \frac{2 x^{\frac{3}{2}}}{3} \) from \( \displaystyle x = 0 \) to \( \displaystyle x = 1 \).
Problem 5.110medium✓ Nihil obstat
Find the length of the curve \( \displaystyle y = \frac{2 \left(x + 2\right)^{\frac{3}{2}}}{3} \) from \( \displaystyle x = 1 \) to \( \displaystyle x = 3 \).
Problem 5.112medium✓ Nihil obstat
Find the length of the curve \( \displaystyle y = \frac{2 \left(x + 1\right)^{\frac{3}{2}}}{3} \) from \( \displaystyle x = 1 \) to \( \displaystyle x = 2 \).
Problem 5.114medium✓ Every equation proved
Find the length of the curve \( \displaystyle y = \frac{2 \left(x + 2\right)^{\frac{3}{2}}}{3} \) from \( \displaystyle x = 0 \) to \( \displaystyle x = 3 \).
Problem 5.115medium✓ Nihil obstat
Find the length of the curve \( \displaystyle y = \frac{2 \left(x + 1\right)^{\frac{3}{2}}}{3} \) from \( \displaystyle x = 0 \) to \( \displaystyle x = 2 \).
Problem 5.116medium✓ Every equation proved
Find the length of the curve \( \displaystyle y = \frac{2 \left(x + 1\right)^{\frac{3}{2}}}{3} \) from \( \displaystyle x = 2 \) to \( \displaystyle x = 4 \).
Problem 5.117medium✓ Nihil obstat
Find the length of the curve \( \displaystyle y = \frac{2 x^{\frac{3}{2}}}{3} \) from \( \displaystyle x = 2 \) to \( \displaystyle x = 5 \).
Problem 5.134medium✓ Nihil obstat
Find the length of the curve \( \displaystyle y = \frac{2 \left(x + 3\right)^{\frac{3}{2}}}{3} \) from \( \displaystyle x = 0 \) to \( \displaystyle x = 1 \).
Problem 5.135medium✓ Nihil obstat
Find the length of the curve \( \displaystyle y = \frac{2 \left(x + 3\right)^{\frac{3}{2}}}{3} \) from \( \displaystyle x = 2 \) to \( \displaystyle x = 5 \).
Problem 5.137medium✓ Nihil obstat
Find the length of the curve \( \displaystyle y = \frac{2 \left(x + 1\right)^{\frac{3}{2}}}{3} \) from \( \displaystyle x = 2 \) to \( \displaystyle x = 3 \).
Problem 5.138medium✓ Nihil obstat
Find the length of the curve \( \displaystyle y = \frac{2 \left(x + 1\right)^{\frac{3}{2}}}{3} \) from \( \displaystyle x = 2 \) to \( \displaystyle x = 9 \).
Problem 5.140medium✓ Nihil obstat
Find the length of the curve \( \displaystyle y = \frac{2 \left(x + 3\right)^{\frac{3}{2}}}{3} \) from \( \displaystyle x = 0 \) to \( \displaystyle x = 3 \).
Problem 5.141medium✓ Nihil obstat
Find the length of the curve \( \displaystyle y = \frac{2 \left(x + 2\right)^{\frac{3}{2}}}{3} \) from \( \displaystyle x = 1 \) to \( \displaystyle x = 6 \).
Problem 5.159medium✓ Nihil obstat
Find the length of the curve \( \displaystyle y = \frac{2 x^{\frac{3}{2}}}{3} \) from \( \displaystyle x = 1 \) to \( \displaystyle x = 3 \).
Problem 5.161medium✓ Every equation proved
Find the length of the curve \( \displaystyle y = \frac{2 x^{\frac{3}{2}}}{3} \) from \( \displaystyle x = 0 \) to \( \displaystyle x = 2 \).
Problem 5.163medium✓ Every equation proved
Find the length of the curve \( \displaystyle y = \frac{2 \left(x + 2\right)^{\frac{3}{2}}}{3} \) from \( \displaystyle x = 1 \) to \( \displaystyle x = 8 \).
Problem 5.165medium✓ Nihil obstat
Find the length of the curve \( \displaystyle y = \frac{2 x^{\frac{3}{2}}}{3} \) from \( \displaystyle x = 0 \) to \( \displaystyle x = 7 \).
Problem 5.182medium✓ Every equation proved
Find the length of the curve \( \displaystyle y = \frac{2 \left(x + 3\right)^{\frac{3}{2}}}{3} \) from \( \displaystyle x = 0 \) to \( \displaystyle x = 2 \).
Problem 5.183medium✓ Every equation proved
Find the length of the curve \( \displaystyle y = \frac{2 \left(x + 3\right)^{\frac{3}{2}}}{3} \) from \( \displaystyle x = 0 \) to \( \displaystyle x = 7 \).
Problem 5.184medium✓ Nihil obstat
Find the length of the curve \( \displaystyle y = \frac{2 \left(x + 3\right)^{\frac{3}{2}}}{3} \) from \( \displaystyle x = 0 \) to \( \displaystyle x = 5 \).
Problem 5.185medium✓ Nihil obstat
Find the length of the curve \( \displaystyle y = \frac{2 \left(x + 2\right)^{\frac{3}{2}}}{3} \) from \( \displaystyle x = 0 \) to \( \displaystyle x = 1 \).
Problem 5.188medium✓ Nihil obstat
Find the length of the curve \( \displaystyle y = \frac{2 x^{\frac{3}{2}}}{3} \) from \( \displaystyle x = 1 \) to \( \displaystyle x = 8 \).
Problem 5.199medium✓ Every equation proved
Find the length of the curve \( \displaystyle y = \frac{2 \left(x + 2\right)^{\frac{3}{2}}}{3} \) from \( \displaystyle x = 1 \) to \( \displaystyle x = 4 \).
Problem 5.200medium✓ Nihil obstat
Find the length of the curve \( \displaystyle y = \frac{2 x^{\frac{3}{2}}}{3} \) from \( \displaystyle x = 0 \) to \( \displaystyle x = 5 \).
Problem 5.816medium✓ Nihil obstat
Find the length of the curve \( \displaystyle y = \frac{2 \left(x + 3\right)^{\frac{3}{2}}}{3} \) from \( \displaystyle x = 2 \) to \( \displaystyle x = 3 \).
Problem 5.46hard✓ Every equation proved
Find the length of the curve \( \displaystyle y = \frac{2 x^{\frac{3}{2}}}{3} \) from \( \displaystyle x = 1 \) to \( \displaystyle x = 6 \).
Problem 5.47hard✓ Every equation proved
Find the length of the curve \( \displaystyle y = \frac{2 \left(x + 2\right)^{\frac{3}{2}}}{3} \) from \( \displaystyle x = 2 \) to \( \displaystyle x = 9 \).
Problem 5.52hard✓ Nihil obstat
Find the length of the curve \( \displaystyle y = \frac{2 \left(x + 3\right)^{\frac{3}{2}}}{3} \) from \( \displaystyle x = 1 \) to \( \displaystyle x = 3 \).
Problem 5.55hard✓ Nihil obstat
Find the length of the curve \( \displaystyle y = \frac{2 \left(x + 3\right)^{\frac{3}{2}}}{3} \) from \( \displaystyle x = 1 \) to \( \displaystyle x = 4 \).
Problem 5.56hard✓ Nihil obstat
Find the length of the curve \( \displaystyle y = \frac{2 \left(x + 2\right)^{\frac{3}{2}}}{3} \) from \( \displaystyle x = 0 \) to \( \displaystyle x = 7 \).
Problem 5.59hard✓ Nihil obstat
Find the length of the curve \( \displaystyle y = \frac{2 \left(x + 1\right)^{\frac{3}{2}}}{3} \) from \( \displaystyle x = 1 \) to \( \displaystyle x = 3 \).
Problem 5.111hard✓ Nihil obstat
Find the length of the curve \( \displaystyle y = \frac{2 x^{\frac{3}{2}}}{3} \) from \( \displaystyle x = 2 \) to \( \displaystyle x = 9 \).
Problem 5.113hard✓ Nihil obstat
Find the length of the curve \( \displaystyle y = \frac{2 \left(x + 3\right)^{\frac{3}{2}}}{3} \) from \( \displaystyle x = 2 \) to \( \displaystyle x = 7 \).
Problem 5.136hard✓ Nihil obstat
Find the length of the curve \( \displaystyle y = \frac{2 \left(x + 1\right)^{\frac{3}{2}}}{3} \) from \( \displaystyle x = 0 \) to \( \displaystyle x = 5 \).
Problem 5.139hard✓ Nihil obstat
Find the length of the curve \( \displaystyle y = \frac{2 \left(x + 3\right)^{\frac{3}{2}}}{3} \) from \( \displaystyle x = 2 \) to \( \displaystyle x = 9 \).
Problem 5.158hard✓ Every equation proved
Find the length of the curve \( \displaystyle y = \frac{2 \left(x + 1\right)^{\frac{3}{2}}}{3} \) from \( \displaystyle x = 1 \) to \( \displaystyle x = 6 \).
Problem 5.160hard✓ Every equation proved
Find the length of the curve \( \displaystyle y = \frac{2 \left(x + 1\right)^{\frac{3}{2}}}{3} \) from \( \displaystyle x = 0 \) to \( \displaystyle x = 3 \).
Problem 5.162hard✓ Nihil obstat
Find the length of the curve \( \displaystyle y = \frac{2 \left(x + 2\right)^{\frac{3}{2}}}{3} \) from \( \displaystyle x = 2 \) to \( \displaystyle x = 5 \).
Problem 5.164hard✓ Nihil obstat
Find the length of the curve \( \displaystyle y = \frac{2 \left(x + 2\right)^{\frac{3}{2}}}{3} \) from \( \displaystyle x = 2 \) to \( \displaystyle x = 7 \).
Problem 5.186hard✓ Nihil obstat
Find the length of the curve \( \displaystyle y = \frac{2 \left(x + 3\right)^{\frac{3}{2}}}{3} \) from \( \displaystyle x = 1 \) to \( \displaystyle x = 6 \).
Problem 5.187hard✓ Every equation proved
Find the length of the curve \( \displaystyle y = \frac{2 x^{\frac{3}{2}}}{3} \) from \( \displaystyle x = 2 \) to \( \displaystyle x = 7 \).
Problem 5.189hard✓ Nihil obstat
Find the length of the curve \( \displaystyle y = \frac{2 \left(x + 2\right)^{\frac{3}{2}}}{3} \) from \( \displaystyle x = 2 \) to \( \displaystyle x = 3 \).
Problem 5.201hard✓ Nihil obstat
Find the length of the curve \( \displaystyle y = \frac{2 \left(x + 2\right)^{\frac{3}{2}}}{3} \) from \( \displaystyle x = 0 \) to \( \displaystyle x = 2 \).
Problem 5.202hard✓ Nihil obstat
Find the length of the curve \( \displaystyle y = \frac{2 \left(x + 3\right)^{\frac{3}{2}}}{3} \) from \( \displaystyle x = 2 \) to \( \displaystyle x = 4 \).
Problem 5.813hard✓ Nihil obstat
Find the length of the curve \( \displaystyle y = \frac{2 \left(x + 1\right)^{\frac{3}{2}}}{3} \) from \( \displaystyle x = 1 \) to \( \displaystyle x = 8 \).
Problem 5.814hard✓ Nihil obstat
Find the length of the curve \( \displaystyle y = \frac{2 \left(x + 1\right)^{\frac{3}{2}}}{3} \) from \( \displaystyle x = 0 \) to \( \displaystyle x = 1 \).
Problem 5.815hard✓ Nihil obstat
Find the length of the curve \( \displaystyle y = \frac{2 x^{\frac{3}{2}}}{3} \) from \( \displaystyle x = 1 \) to \( \displaystyle x = 4 \).
Problem 5.817hard✓ Nihil obstat
Find the length of the curve \( \displaystyle y = \frac{2 x^{\frac{3}{2}}}{3} \) from \( \displaystyle x = 1 \) to \( \displaystyle x = 2 \).
Problem 5.818hard✓ Nihil obstat