11.3: Line Integrals
- Page ID
- 144359
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- Evaluate \(\displaystyle \int_C xy^2\ ds\), where \(C\) is the line segment from \((1, 2, 0)\) to \((2, 1, 3)\).
- Evaluate \(\displaystyle \int_C (x + y)\ ds\), where \(C\) is the line segment from \((0, 1, 0)\) to \((1, 0, 0)\).
- Evaluate \(\displaystyle \int_C x^2yz\ ds\), where \(C\) is the curve \(\mathbf{r}(t) = (6t^3, 3t^2, t)\), \(0 \leq t \leq 1\).
- Evaluate \(\displaystyle \int_C \dfrac{y}{1 + x^2}\ ds\), where \(C\) is the curve \(\mathbf{r}(t) = t\mathbf{i} + (1 + 2t)\mathbf{j}\), \(0 \leq t \leq 1\).
- Evaluate \(\displaystyle \int_C \mathbf{F}\cdot d\mathbf{r}\), where \(\mathbf{F}(x, y) = (\sin x, \cos y)\), and \(C\) is the top half of the unit circle counterclockwise from \((1, 0)\) to \((-1, 0)\).
- Evaluate \(\displaystyle \int_C \mathbf{F}\cdot d\mathbf{r}\), where \(\mathbf{F}(x, y) = (\sin x, \cos y)\), and \(C\) is the top half of the unit circle clockwise from \((-1, 0)\) to \((1, 0)\).
- Evaluate \(\displaystyle \int_C (xe^y, x^2y) \cdot d\mathbf{r}\), where \(C\) is the curve \(\mathbf{r}(t) = (3t, t^2)\), \(0 \leq t \leq 1\).
- Evaluate \(\displaystyle \int_C \dfrac{1}{xy}\ dx + \dfrac{1}{x + y}\ dy\), where \(C\) is the path from \((1, 1)\) to \((3, 1)\) to \((3, 6)\).
- Evaluate \(\displaystyle \int_C x^2yz\ dx\), where \(C\) is the curve \(\mathbf{r}(t) = (6t^3, 3t^2, t)\), \(0 \leq t \leq 1\).
- Find the work done by the force field \(\mathbf{F}(x, y, z) = x\mathbf{i} + 3xy\mathbf{j} - (x + z)\mathbf{k}\) on a particle moving along a line segment that goes from \((1, 4, 2)\) to \((0, 5, 1)\).
- Find the work done by the force field \(\mathbf{F}(x, y, z) = (y, z, x)\) on an object that moves along the curve \(\mathbf{r}(t) = \left(\sqrt{t}, \dfrac{1}{\sqrt{t}}, t\right)\), \(1 \leq t \leq 4\).
- Compute the circulation of \(\mathbf{F}(x, y) = (-2, y)\) along the circle of radius \(2\) centered at the origin, oriented counterclockwise.
- Compute the flux of \(\mathbf{F}(x, y) = x^2\mathbf{i} + y\mathbf{j}\) across a line segment from \((0, 0)\) to \((1, 2)\).
- Compute the flux of \(\mathbf{F}(x, y) = (-y, x)\) across the unit circle centered at the origin, oriented counterclockwise.