Exercise · ZW-SCALAR-LINE-INTEGRAL-008 · ★★☆ Intermediate
Compute \(\int_C x\,ds\) on the parabola \(\mathbf r(t)=\langle t,t^2\rangle\), \(0\le t\le1\).
Choose one option.
Show answer
- C. \(\tfrac{5\sqrt5-1}{12}\)
Explanation
\(\|\mathbf r'\|=\sqrt{1+4t^2}\) and with \(u=1+4t^2\) we get \(\int_0^1 t\sqrt{1+4t^2}\,dt=\tfrac1{12}(5\sqrt5-1)\).
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-009 Compute ∫_C z ds on the helix mathbf r(t)=langlecos t,sin t,trangle, 0≤ t≤2π.