∫x² ds along a parabola

∫C (x^2) ds along r(t) = (t, t^2, 0) equals 0.60633731. A worked scalar field line integral with an editable calculator.

∫C (x^2) ds = 0.60633731. Here |r′(t)| = √(1 + 4t²) varies along the curve, which is why the answer is not a round number.

Example: x+y, x^2*y, sqrt(x^2+y^2). Use f = 1 to get arc length.

Curve C: r(t) = (x(t), y(t), z(t))

Leave z(t) as 0 for a plane curve

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The setup

∫C (x^2) ds = 0.60633731

  • Curve: r(t) = (t, t^2, 0) for t from 0 to 1
  • Arc length of the curve: 1.478943
  • Type: scalar field line integral

Here |r′(t)| = √(1 + 4t²) varies along the curve, which is why the answer is not a round number.

More worked line integrals

Work done by F = (y, x) around the unit circle This field is conservative (it is the gradient of xy), so the work around any closed loop is zero. Circulation of F = (−y, x) around a circle of radius 3 Green's theorem predicts 2 × area = 2π(3²) = 18π; the calculator confirms it. Arc length of a helix Setting f = 1 turns the scalar line integral into arc length: 2π√2 for one turn. ∫(x + y) ds along a line segment For a straight parametrisation the speed |r′(t)| is constant, so it factors out of the integral. Work done by F = (x, y, z) along a straight path A radial field along a straight line: the work is simply the change in ½|r|². Circumference of a circle as a line integral A radius-5 circle has circumference 10π — a good sanity check on the parametrisation. Work against a constant force along a curve A constant downward force does work that depends only on the height change, not the path taken.

Frequently asked questions

What is the value of this line integral? ▾

∫C (x^2) ds = 0.60633731. Here |r′(t)| = √(1 + 4t²) varies along the curve, which is why the answer is not a round number.

What curve is being integrated along? ▾

The curve is parametrised as r(t) = (t, t^2, 0) for t from 0 to 1, and it has arc length 1.478943.

Does the direction of travel matter here? ▾

No. A scalar line integral is taken with respect to arc length, so travelling the curve backwards gives exactly the same value.

About this Calculator

∫C (x^2) ds along r(t) = (t, t^2, 0) equals 0.60633731. A worked scalar field line integral with an editable calculator.

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