Exercise · ZW-SCALAR-LINE-INTEGRAL-010 · ★★★ Advanced
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 \(2\)).
Choose one option.
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- D. \(\pi\)
Explanation
\(\|\mathbf r'\|=2\) and \(\int_0^\pi 2\cos^2 2t\,dt=\pi\). It is the same value as with the standard parametrization, since the integral depends on the curve, not on the speed.
Review the concept: Line integral of a scalar function
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 \(ds\).
\[\int_C f\,ds=\int_a^b f(\mathbf r(t))\,\|\mathbf r'(t)\|\,dt\]
More practice problems: line integral of a scalar function
- ZW-SCALAR-LINE-INTEGRAL-001 Compute ∫_C 2 ds, with C the segment mathbf r(t)=langle 3t,4trangle, 0≤ t≤1.
- ZW-SCALAR-LINE-INTEGRAL-002 Compute ∫_C x ds, with mathbf r(t)=langle t,2trangle, 0≤ t≤1.
- ZW-SCALAR-LINE-INTEGRAL-003 Compute ∫_C xy ds over the arc mathbf r(t)=langlecos t,sin trangle, 0≤ t≤π/2.
- ZW-SCALAR-LINE-INTEGRAL-004 Compute ∫_C (x^2+y^2) ds on the circle mathbf r(t)=langle 3cos t,3sin trangle, 0≤ t≤2π.
- ZW-SCALAR-LINE-INTEGRAL-005 If the orientation of C is reversed, what happens to ∫_C f ds?
- ZW-SCALAR-LINE-INTEGRAL-006 Compute the length of the curve mathbf r(t)=langle 2t,3t,6trangle, 0≤ t≤1, as ∫_C 1 ds.
- ZW-SCALAR-LINE-INTEGRAL-007 Compute ∫_C y ds on the upper semicircle mathbf r(t)=langlecos t,sin trangle, 0≤ t≤π.
- ZW-SCALAR-LINE-INTEGRAL-008 Compute ∫_C x ds on the parabola mathbf r(t)=langle t,t^2rangle, 0≤ t≤1.