Chapter 13: Problem 48
Evaluate the line integral along the path \(C\) given by \(x=2 t, y=10 t,\) where \(0 \leq t \leq 1\) \(\int_{C}(3 y-x) d x+y^{2} d y\)
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Evaluate \(\int_{C} \mathbf{F} \cdot d \mathbf{r}\) for each curve. Discuss the orientation of the curve and its effect on the value of the integral. \(\mathbf{F}(x, y)=x^{2} y \mathbf{i}+x y^{3 / 2} \mathbf{j}\) (a) \(\mathbf{r}_{1}(t)=(t+1) \mathbf{i}+t^{2} \mathbf{j}, \quad 0 \leq t \leq 2\) (b) \(\mathbf{r}_{2}(t)=(1+2 \cos t) \mathbf{i}+\left(4 \cos ^{2} t\right) \mathbf{j}, \quad 0 \leq t \leq \pi / 2\)
In Exercises \(5-8,\) evaluate the line integral along the given path. \(\int_{C}(x-y) d s\) \(C: \mathbf{r}(t)=4 t \mathbf{i}+3 t \mathbf{j}\) \(0 \leq t \leq 2\)
Evaluate \(\int_{C} \mathbf{F} \cdot d \mathbf{r}\) where \(C\) is represented by \(\mathbf{r}(t)\) \(\mathbf{F}(x, y)=3 x \mathbf{i}+4 y \mathbf{j}\) \(\quad C: \mathbf{r}(t)=2 \cos t \mathbf{i}+2 \sin t \mathbf{j}, \quad 0 \leq t \leq \pi / 2\)
Determine the value of \(c\) such that the work done by the force field \(\mathbf{F}(x, y)=15\left[\left(4-x^{2} y\right) \mathbf{i}-x y \mathbf{j}\right]\) on an object moving along the parabolic path \(y=c\left(1-x^{2}\right)\) between the points (-1,0) and (1,0) is a minimum. Compare the result with the work required to move the object along the straight-line path connecting the points.
In Exercises \(43-46,\) demonstrate the property that \(\int_{C} \mathbf{F} \cdot d \mathbf{r}=\mathbf{0}\) regardless of the initial and terminal points of \(C,\) if the tangent vector \(\mathbf{r}^{\prime}(t)\) is orthogonal to the force field \(\mathbf{F}\) \(\mathbf{F}(x, y)=y \mathbf{i}-x \mathbf{j}\) \(C: \mathbf{r}(t)=t \mathbf{i}-2 t \mathbf{j}\)
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