Line and surface integrals · Definition

Line integral of a scalar function

multivariable calculusline integrals

Before you start you need to know

It extends the one-variable integral to a curve in the plane or space: it gives masses of wires, charges and averages along paths.

Definition

The line integral of a scalar function \(f\) over a curve \(C\) parametrized by \(\mathbf r(t)\), \(a\le t\le b\), accumulates the values of \(f\) weighted by the arc length \(ds\).

\[\int_C f\,ds=\int_a^b f(\mathbf r(t))\,\|\mathbf r'(t)\|\,dt\]

With \(f=1\) the line integral is the length of the curve, \(\int_C ds=L\). If \(\rho\) is the linear density of a wire shaped like \(C\), then \(\int_C\rho\,ds\) is its mass, and \(\tfrac1L\int_C f\,ds\) is the average value of \(f\) over the curve. The factor \(\|\mathbf r'(t)\|\,dt\) is precisely \(ds\), the arc-length element.

To compute it, parametrize \(C\), substitute \(x=x(t)\), \(y=y(t)\), \(z=z(t)\) in \(f\), multiply by \(\|\mathbf r'(t)\|\) and integrate in \(t\). For example, on the unit circle \(\mathbf r=\langle\cos t,\sin t\rangle\) we have \(\|\mathbf r'\|=1\) and \(\int_C y\,ds=\int_0^{\pi}\sin t\,dt=2\) on the upper half. If \(C\) consists of several pieces, add the integrals over each piece.

The result depends neither on the parametrization nor on the orientation: traversing the curve backwards or at another speed changes \(\mathbf r'\), but the length \(ds\) stays positive and the value is the same. This is the key difference from the line integral of a vector field, which does change sign when the direction is reversed.

A typical mistake is to forget the factor \(\|\mathbf r'(t)\|\) and compute \(\int_a^b f(\mathbf r(t))\,dt\), which depends on the chosen parametrization and does not measure the curve. It connects with arc length (the case \(f=1\)) and prepares the line integral of a vector field.

Think about it

What is the mass of a wire shaped like a segment of length \(4\) with constant density \(3\)?

\(\int_C 3\,ds=3\cdot4=12\).

Why is \(\int_a^b f(\mathbf r(t))\,dt\) not a line integral?

It lacks \(\|\mathbf r'(t)\|\); without it the value depends on the speed of the parametrization, not on the curve.

Exercises 10

  1. Compute \(\int_C 2\,ds\), with \(C\) the segment \(\mathbf r(t)=\langle 3t,4t\rangle\), \(0\le t\le1\).
  2. Compute \(\int_C x\,ds\), with \(\mathbf r(t)=\langle t,2t\rangle\), \(0\le t\le1\).
  3. Compute \(\int_C xy\,ds\) over the arc \(\mathbf r(t)=\langle\cos t,\sin t\rangle\), \(0\le t\le\pi/2\).
  4. Compute \(\int_C (x^2+y^2)\,ds\) on the circle \(\mathbf r(t)=\langle 3\cos t,3\sin t\rangle\), \(0\le t\le2\pi\).
  5. If the orientation of \(C\) is reversed, what happens to \(\int_C f\,ds\)?
  6. Compute the length of the curve \(\mathbf r(t)=\langle 2t,3t,6t\rangle\), \(0\le t\le1\), as \(\int_C 1\,ds\).
  7. Compute \(\int_C y\,ds\) on the upper semicircle \(\mathbf r(t)=\langle\cos t,\sin t\rangle\), \(0\le t\le\pi\).
  8. Compute \(\int_C x\,ds\) on the parabola \(\mathbf r(t)=\langle t,t^2\rangle\), \(0\le t\le1\).
  9. Compute \(\int_C z\,ds\) on the helix \(\mathbf r(t)=\langle\cos t,\sin t,t\rangle\), \(0\le t\le2\pi\).
  10. Compute \(\int_C x^2\,ds\) with \(\mathbf r(t)=\langle\cos 2t,\sin 2t\rangle\), \(0\le t\le\pi\) (the circle traversed at speed \(2\)).

References

  • James Stewart, Daniel K. Clegg, Saleem Watson, Calculus: Early Transcendentals, 9th edition, Cengage, 2021. §16.2
  • Joel Hass, Christopher Heil, Przemyslaw Bogacki, Maurice D. Weir, George B. Thomas Jr., Thomas' Calculus, 15th edition, Pearson, 2023. §16.1
  • Jerrold E. Marsden, Anthony Tromba, Vector Calculus, 6th edition, W. H. Freeman, 2012. §7.1