Exercise · ZW-VECTOR-LINE-INTEGRAL-001 · ★☆☆ Basic
Compute \(\int_C \mathbf F\cdot d\mathbf r\) with \(\mathbf F=\langle y,x\rangle\) and \(\mathbf r(t)=\langle t,t^2\rangle\), \(0\le t\le1\).
Choose one option.
Show answer
- C. \(1\)
Explanation
Here \(\mathbf F(\mathbf r(t))=\langle t^2,t\rangle\) and \(\mathbf r'=\langle1,2t\rangle\), so the dot product is \(t^2+2t^2=3t^2\) and the integral is \(1\). Forgetting \(\mathbf r'\) gives \(\tfrac56\).
Review the concept: Line integral of a vector field
The line integral of a vector field \(\mathbf F\) over an oriented curve \(C\), parametrized by \(\mathbf r(t)\), \(a\le t\le b\), measures the work \(\mathbf F\) does along \(C\).
\[\int_C \mathbf F\cdot d\mathbf r=\int_a^b \mathbf F(\mathbf r(t))\cdot\mathbf r'(t)\,dt\]
More practice problems: line integral of a vector field
- ZW-VECTOR-LINE-INTEGRAL-002 Compute ∫_C mathbf F· dmathbf r with mathbf F=langle -y,xrangle over the unit circle traversed once…
- ZW-VECTOR-LINE-INTEGRAL-003 Compute ∫_C mathbf F· dmathbf r with mathbf F=langle x,y,zrangle and mathbf r(t)=langle t,t,trangle…
- ZW-VECTOR-LINE-INTEGRAL-004 If ∫_C mathbf F· dmathbf r=7, what is the integral over the same curve traversed in the opposite di…
- ZW-VECTOR-LINE-INTEGRAL-005 A constant force mathbf F=langle2,3rangle moves a particle in a straight line from (0,0) to (4,1).…
- ZW-VECTOR-LINE-INTEGRAL-006 Compute ∫_C mathbf F· dmathbf r with mathbf F=langle xy, xrangle and mathbf r(t)=langle t,t^2rangle…
- ZW-VECTOR-LINE-INTEGRAL-007 Compute ∫_C mathbf F· dmathbf r with mathbf F=langle z,x,yrangle and mathbf r(t)=langle t,t^2,t^3ra…
- ZW-VECTOR-LINE-INTEGRAL-008 A field mathbf F is perpendicular to the curve C at every point. What is ∫_C mathbf F· dmathbf r?
- ZW-VECTOR-LINE-INTEGRAL-009 The same curve C is traversed with the same orientation, but with a parametrization twice as fast.…