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Line integrals and work practice problems

∫ F · dr along a path: parametrize, dot, integrate. 53 problems with worked solutions; in 53 of them every equation is proved by a computer algebra system.

Interactive: Line Integral Explorer

12 worked problems

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

Problem 12.2

medium
Find the work done by \( \displaystyle \mathbf{F} = \langle - y, x \rangle \) along the unit circle, counterclockwise.
  1. Parametrize: r(t) = ⟨cos(t), sin(t)⟩ for 0 ≤ t ≤ 2*pi.
    Reviewed
  2. \[ \left[\begin{matrix}\frac{d}{d t} \cos{\left(t \right)}\\\frac{d}{d t} \sin{\left(t \right)}\end{matrix}\right] = \left[\begin{matrix}- \sin{\left(t \right)}\\\cos{\left(t \right)}\end{matrix}\right] \]
    r'(t).✓ Proved
  3. \[ \sin^{2}{\left(t \right)} + \cos^{2}{\left(t \right)} = 1 \]
    F(r(t)) · r'(t).✓ Proved
  4. \[ \int\limits_{0}^{2 \pi} 1\, dt = 2 \pi \]
    W = ∫ F · dr.✓ Proved
Answer \( 2 \pi \)

✓ 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 12.3

medium
Find the work done by \( \displaystyle \mathbf{F} = \langle y^{2}, 2 x \rangle \) along the line segment from (0, 0) to (1, 1).
  1. Parametrize: r(t) = ⟨t, t⟩ for 0 ≤ t ≤ 1.
    Reviewed
  2. \[ \left[\begin{matrix}\frac{d}{d t} t\\\frac{d}{d t} t\end{matrix}\right] = \left[\begin{matrix}1\\1\end{matrix}\right] \]
    r'(t).✓ Proved
  3. \[ t^{2} + 2 t = t \left(t + 2\right) \]
    F(r(t)) · r'(t).✓ Proved
  4. \[ \int\limits_{0}^{1} t \left(t + 2\right)\, dt = \frac{4}{3} \]
    W = ∫ F · dr.✓ Proved
Answer \( \frac{4}{3} \)

✓ 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 12.4

medium
Find the work done by \( \displaystyle \mathbf{F} = \langle y^{2}, 2 x \rangle \) along the parabola y = x² from (0, 0) to (1, 1).
  1. Parametrize: r(t) = ⟨t, t**2⟩ for 0 ≤ t ≤ 1.
    Reviewed
  2. \[ \left[\begin{matrix}\frac{d}{d t} t\\\frac{d}{d t} t^{2}\end{matrix}\right] = \left[\begin{matrix}1\\2 t\end{matrix}\right] \]
    r'(t).✓ Proved
  3. \[ t^{4} + 4 t^{2} = t^{2} \left(t^{2} + 4\right) \]
    F(r(t)) · r'(t).✓ Proved
  4. \[ \int\limits_{0}^{1} t^{2} \left(t^{2} + 4\right)\, dt = \frac{23}{15} \]
    W = ∫ F · dr.✓ Proved
Answer \( \frac{23}{15} \)

✓ 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 12.7

medium
Find the work done by \( \displaystyle \mathbf{F} = \langle x^{2}, x y \rangle \) along the parabola y = x² from (0, 0) to (1, 1).
  1. Parametrize: r(t) = ⟨t, t**2⟩ for 0 ≤ t ≤ 1.
    Reviewed
  2. \[ \left[\begin{matrix}\frac{d}{d t} t\\\frac{d}{d t} t^{2}\end{matrix}\right] = \left[\begin{matrix}1\\2 t\end{matrix}\right] \]
    r'(t).✓ Proved
  3. \[ 2 t^{4} + t^{2} \]
    F(r(t)) · r'(t).Reviewed
  4. \[ \int\limits_{0}^{1} \left(2 t^{4} + t^{2}\right)\, dt = \frac{11}{15} \]
    W = ∫ F · dr.✓ Proved
Answer \( \frac{11}{15} \)

✓ 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 12.8

medium
Find the work done by \( \displaystyle \mathbf{F} = \langle - y, x \rangle \) along the line segment from (0, 0) to (2, 3).
  1. Parametrize: r(t) = ⟨2*t, 3*t⟩ for 0 ≤ t ≤ 1.
    Reviewed
  2. \[ \left[\begin{matrix}\frac{d}{d t} 2 t\\\frac{d}{d t} 3 t\end{matrix}\right] = \left[\begin{matrix}2\\3\end{matrix}\right] \]
    r'(t).✓ Proved
  3. \[ 0 \]
    F(r(t)) · r'(t).Reviewed
  4. \[ \int\limits_{0}^{1} 0\, dt = 0 \]
    W = ∫ F · dr.✓ Proved
Answer \( 0 \)

✓ 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 12.9

medium
Find the work done by \( \displaystyle \mathbf{F} = \langle y, x \rangle \) along the parabola y = x² from (0, 0) to (1, 1).
  1. Parametrize: r(t) = ⟨t, t**2⟩ for 0 ≤ t ≤ 1.
    Reviewed
  2. \[ \left[\begin{matrix}\frac{d}{d t} t\\\frac{d}{d t} t^{2}\end{matrix}\right] = \left[\begin{matrix}1\\2 t\end{matrix}\right] \]
    r'(t).✓ Proved
  3. \[ 3 t^{2} \]
    F(r(t)) · r'(t).Reviewed
  4. \[ \int\limits_{0}^{1} 3 t^{2}\, dt = 1 \]
    W = ∫ F · dr.✓ Proved
Answer \( 1 \)

✓ 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 12.12

medium
Find the work done by \( \displaystyle \mathbf{F} = \langle x^{2}, x y \rangle \) along the line segment from (0, 0) to (2, 3).
  1. Parametrize: r(t) = ⟨2*t, 3*t⟩ for 0 ≤ t ≤ 1.
    Reviewed
  2. \[ \left[\begin{matrix}\frac{d}{d t} 2 t\\\frac{d}{d t} 3 t\end{matrix}\right] = \left[\begin{matrix}2\\3\end{matrix}\right] \]
    r'(t).✓ Proved
  3. \[ 26 t^{2} \]
    F(r(t)) · r'(t).Reviewed
  4. \[ \int\limits_{0}^{1} 26 t^{2}\, dt = \frac{26}{3} \]
    W = ∫ F · dr.✓ Proved
Answer \( \frac{26}{3} \)

✓ 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 12.13

medium
Find the work done by \( \displaystyle \mathbf{F} = \langle y, x \rangle \) along the line segment from (0, 0) to (1, 3).
  1. Parametrize: r(t) = ⟨t, 3*t⟩ for 0 ≤ t ≤ 1.
    Reviewed
  2. \[ \left[\begin{matrix}\frac{d}{d t} t\\\frac{d}{d t} 3 t\end{matrix}\right] = \left[\begin{matrix}1\\3\end{matrix}\right] \]
    r'(t).✓ Proved
  3. \[ 6 t \]
    F(r(t)) · r'(t).Reviewed
  4. \[ \int\limits_{0}^{1} 6 t\, dt = 3 \]
    W = ∫ F · dr.✓ Proved
Answer \( 3 \)

✓ 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 12.15

medium
Find the work done by \( \displaystyle \mathbf{F} = \langle x + y, x - y \rangle \) along the unit circle, counterclockwise.
  1. Parametrize: r(t) = ⟨cos(t), sin(t)⟩ for 0 ≤ t ≤ 2*pi.
    Reviewed
  2. \[ \left[\begin{matrix}\frac{d}{d t} \cos{\left(t \right)}\\\frac{d}{d t} \sin{\left(t \right)}\end{matrix}\right] = \left[\begin{matrix}- \sin{\left(t \right)}\\\cos{\left(t \right)}\end{matrix}\right] \]
    r'(t).✓ Proved
  3. \[ \left(- \sin{\left(t \right)} + \cos{\left(t \right)}\right) \cos{\left(t \right)} - \left(\sin{\left(t \right)} + \cos{\left(t \right)}\right) \sin{\left(t \right)} = \sqrt{2} \cos{\left(2 t + \frac{\pi}{4} \right)} \]
    F(r(t)) · r'(t).✓ Proved
  4. \[ \int\limits_{0}^{2 \pi} \sqrt{2} \cos{\left(2 t + \frac{\pi}{4} \right)}\, dt = 0 \]
    W = ∫ F · dr.✓ Proved
Answer \( 0 \)

✓ 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 12.37

medium
Find the work done by \( \displaystyle \mathbf{F} = \langle y^{2}, 2 x \rangle \) along the unit circle, counterclockwise.
  1. Parametrize: r(t) = ⟨cos(t), sin(t)⟩ for 0 ≤ t ≤ 2*pi.
    Reviewed
  2. \[ \left[\begin{matrix}\frac{d}{d t} \cos{\left(t \right)}\\\frac{d}{d t} \sin{\left(t \right)}\end{matrix}\right] = \left[\begin{matrix}- \sin{\left(t \right)}\\\cos{\left(t \right)}\end{matrix}\right] \]
    r'(t).✓ Proved
  3. \[ - \sin^{3}{\left(t \right)} + 2 \cos^{2}{\left(t \right)} \]
    F(r(t)) · r'(t).Reviewed
  4. \[ \int\limits_{0}^{2 \pi} \left(- \sin^{3}{\left(t \right)} + 2 \cos^{2}{\left(t \right)}\right)\, dt = 2 \pi \]
    W = ∫ F · dr.✓ Proved
Answer \( 2 \pi \)

✓ 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 12.38

medium
Find the work done by \( \displaystyle \mathbf{F} = \langle x + y, x - y \rangle \) along the line segment from (0, 0) to (3, 2).
  1. Parametrize: r(t) = ⟨3*t, 2*t⟩ for 0 ≤ t ≤ 1.
    Reviewed
  2. \[ \left[\begin{matrix}\frac{d}{d t} 3 t\\\frac{d}{d t} 2 t\end{matrix}\right] = \left[\begin{matrix}3\\2\end{matrix}\right] \]
    r'(t).✓ Proved
  3. \[ 17 t \]
    F(r(t)) · r'(t).Reviewed
  4. \[ \int\limits_{0}^{1} 17 t\, dt = \frac{17}{2} \]
    W = ∫ F · dr.✓ Proved
Answer \( \frac{17}{2} \)

✓ 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 12.42

medium
Find the work done by \( \displaystyle \mathbf{F} = \langle y, x \rangle \) along the unit circle, counterclockwise.
  1. Parametrize: r(t) = ⟨cos(t), sin(t)⟩ for 0 ≤ t ≤ 2*pi.
    Reviewed
  2. \[ \left[\begin{matrix}\frac{d}{d t} \cos{\left(t \right)}\\\frac{d}{d t} \sin{\left(t \right)}\end{matrix}\right] = \left[\begin{matrix}- \sin{\left(t \right)}\\\cos{\left(t \right)}\end{matrix}\right] \]
    r'(t).✓ Proved
  3. \[ - \sin^{2}{\left(t \right)} + \cos^{2}{\left(t \right)} = \cos{\left(2 t \right)} \]
    F(r(t)) · r'(t).✓ Proved
  4. \[ \int\limits_{0}^{2 \pi} \cos{\left(2 t \right)}\, dt = 0 \]
    W = ∫ F · dr.✓ Proved
Answer \( 0 \)

✓ 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 line integrals and work practice

Find the work done by \( \displaystyle \mathbf{F} = \langle x^{2}, x y \rangle \) along the unit circle, counterclockwise.
Problem 12.1medium✓ Every equation proved
Find the work done by \( \displaystyle \mathbf{F} = \langle - y, x \rangle \) along the parabola y = x² from (0, 0) to (1, 1).
Problem 12.5medium✓ Every equation proved
Find the work done by \( \displaystyle \mathbf{F} = \langle y^{2}, 2 x \rangle \) along the line segment from (0, 0) to (2, 2).
Problem 12.6medium✓ Nihil obstat
Find the work done by \( \displaystyle \mathbf{F} = \langle x + y, x - y \rangle \) along the parabola y = x² from (0, 0) to (1, 1).
Problem 12.10medium✓ Every equation proved
Find the work done by \( \displaystyle \mathbf{F} = \langle y^{2}, 2 x \rangle \) along the line segment from (0, 0) to (1, 3).
Problem 12.11medium✓ Nihil obstat
Find the work done by \( \displaystyle \mathbf{F} = \langle - y, x \rangle \) along the line segment from (0, 0) to (3, 2).
Problem 12.14medium✓ Nihil obstat
Find the work done by \( \displaystyle \mathbf{F} = \langle y, x \rangle \) along the line segment from (0, 0) to (3, 3).
Problem 12.39medium✓ Nihil obstat
Find the work done by \( \displaystyle \mathbf{F} = \langle x + y, x - y \rangle \) along the line segment from (0, 0) to (2, 2).
Problem 12.40medium✓ Nihil obstat
Find the work done by \( \displaystyle \mathbf{F} = \langle x^{2}, x y \rangle \) along the line segment from (0, 0) to (1, 3).
Problem 12.41medium✓ Nihil obstat
Find the work done by \( \displaystyle \mathbf{F} = \langle x^{2}, x y \rangle \) along the line segment from (0, 0) to (3, 3).
Problem 12.43medium✓ Nihil obstat
Find the work done by \( \displaystyle \mathbf{F} = \langle x + y, x - y \rangle \) along the line segment from (0, 0) to (3, 3).
Problem 12.44medium✓ Nihil obstat
Find the work done by \( \displaystyle \mathbf{F} = \langle x + y, x - y \rangle \) along the line segment from (0, 0) to (1, 3).
Problem 12.45medium✓ Every equation proved
Find the work done by \( \displaystyle \mathbf{F} = \langle y, x \rangle \) along the line segment from (0, 0) to (1, 2).
Problem 12.46medium✓ Nihil obstat
Find the work done by \( \displaystyle \mathbf{F} = \langle x^{2}, x y \rangle \) along the line segment from (0, 0) to (2, 1).
Problem 12.47medium✓ Nihil obstat
Find the work done by \( \displaystyle \mathbf{F} = \langle - y, x \rangle \) along the line segment from (0, 0) to (1, 1).
Problem 12.48medium✓ Nihil obstat
Find the work done by \( \displaystyle \mathbf{F} = \langle x^{2}, x y \rangle \) along the line segment from (0, 0) to (2, 2).
Problem 12.49medium✓ Nihil obstat
Find the work done by \( \displaystyle \mathbf{F} = \langle y^{2}, 2 x \rangle \) along the line segment from (0, 0) to (3, 2).
Problem 12.50medium✓ Nihil obstat
Find the work done by \( \displaystyle \mathbf{F} = \langle y, x \rangle \) along the line segment from (0, 0) to (2, 3).
Problem 12.51medium✓ Nihil obstat
Find the work done by \( \displaystyle \mathbf{F} = \langle y, x \rangle \) along the line segment from (0, 0) to (3, 1).
Problem 12.82medium✓ Nihil obstat
Find the work done by \( \displaystyle \mathbf{F} = \langle x + y, x - y \rangle \) along the line segment from (0, 0) to (1, 1).
Problem 12.83medium✓ Every equation proved
Find the work done by \( \displaystyle \mathbf{F} = \langle x + y, x - y \rangle \) along the line segment from (0, 0) to (1, 2).
Problem 12.84medium✓ Nihil obstat
Find the work done by \( \displaystyle \mathbf{F} = \langle x + y, x - y \rangle \) along the line segment from (0, 0) to (3, 1).
Problem 12.85medium✓ Nihil obstat
Find the work done by \( \displaystyle \mathbf{F} = \langle - y, x \rangle \) along the line segment from (0, 0) to (2, 2).
Problem 12.86medium✓ Nihil obstat
Find the work done by \( \displaystyle \mathbf{F} = \langle - y, x \rangle \) along the line segment from (0, 0) to (1, 2).
Problem 12.87medium✓ Nihil obstat
Find the work done by \( \displaystyle \mathbf{F} = \langle - y, x \rangle \) along the line segment from (0, 0) to (1, 3).
Problem 12.88medium✓ Every equation proved
Find the work done by \( \displaystyle \mathbf{F} = \langle y, x \rangle \) along the line segment from (0, 0) to (2, 1).
Problem 12.89medium✓ Nihil obstat
Find the work done by \( \displaystyle \mathbf{F} = \langle y^{2}, 2 x \rangle \) along the line segment from (0, 0) to (1, 2).
Problem 12.106medium✓ Nihil obstat
Find the work done by \( \displaystyle \mathbf{F} = \langle x^{2}, x y \rangle \) along the line segment from (0, 0) to (3, 1).
Problem 12.107medium✓ Nihil obstat
Find the work done by \( \displaystyle \mathbf{F} = \langle y^{2}, 2 x \rangle \) along the line segment from (0, 0) to (2, 3).
Problem 12.108medium✓ Nihil obstat
Find the work done by \( \displaystyle \mathbf{F} = \langle x^{2}, x y \rangle \) along the line segment from (0, 0) to (3, 2).
Problem 12.109medium✓ Nihil obstat
Find the work done by \( \displaystyle \mathbf{F} = \langle - y, x \rangle \) along the line segment from (0, 0) to (2, 1).
Problem 12.110medium✓ Every equation proved
Find the work done by \( \displaystyle \mathbf{F} = \langle - y, x \rangle \) along the line segment from (0, 0) to (3, 3).
Problem 12.111medium✓ Nihil obstat
Find the work done by \( \displaystyle \mathbf{F} = \langle y^{2}, 2 x \rangle \) along the line segment from (0, 0) to (3, 3).
Problem 12.112medium✓ Nihil obstat
Find the work done by \( \displaystyle \mathbf{F} = \langle x + y, x - y \rangle \) along the line segment from (0, 0) to (2, 3).
Problem 12.113medium✓ Nihil obstat
Find the work done by \( \displaystyle \mathbf{F} = \langle x + y, x - y \rangle \) along the line segment from (0, 0) to (2, 1).
Problem 12.128medium✓ Nihil obstat
Find the work done by \( \displaystyle \mathbf{F} = \langle y, x \rangle \) along the line segment from (0, 0) to (3, 2).
Problem 12.146medium✓ Nihil obstat
Find the work done by \( \displaystyle \mathbf{F} = \langle x^{2}, x y \rangle \) along the line segment from (0, 0) to (1, 1).
Problem 12.147medium✓ Nihil obstat
Find the work done by \( \displaystyle \mathbf{F} = \langle - y, x \rangle \) along the line segment from (0, 0) to (3, 1).
Problem 12.148medium✓ Every equation proved
Find the work done by \( \displaystyle \mathbf{F} = \langle y, x \rangle \) along the line segment from (0, 0) to (2, 2).
Problem 12.149medium✓ Nihil obstat
Find the work done by \( \displaystyle \mathbf{F} = \langle x^{2}, x y \rangle \) along the line segment from (0, 0) to (1, 2).
Problem 12.150medium✓ Nihil obstat
Find the work done by \( \displaystyle \mathbf{F} = \langle y^{2}, 2 x \rangle \) along the line segment from (0, 0) to (3, 1).
Problem 12.151medium✓ Nihil obstat