11.3: Line Integrals
- Page ID
- 144359
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- 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.