Chapter 13: Problem 49
In Exercises \(49-54,\) evaluate the integral \(\int_{C}(2 x-y) d x+(x+3 y) d y\) along the path \(C\). \(C: x\) -axis from \(x=0\) to \(x=5\)
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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 \(\mathbf{F}\) \(\mathbf{F}(x, y)=-3 y \mathbf{i}+x \mathbf{j}\) \(C: \mathbf{r}(t)=t \mathbf{i}-t^{3} \mathbf{j}\)
Consider a wire of density \(\rho(x, y)\) given by the space curve \(C: \mathbf{r}(t)=x(t) \mathbf{i}+y(t) \mathbf{j}, \quad a \leq t \leq b\) The moments of inertia about the \(x\) - and \(y\) -axes are given by \(I_{x}=\int_{C} y^{2} \rho(x, y) d s\) and \(I_{y}=\int_{C} x^{2} \rho(x, y) d s\) Find the moments of inertia for the wire of density \(\boldsymbol{\rho}\). A wire lies along \(\mathbf{r}(t)=a \cos t \mathbf{i}+a \sin t \mathbf{j}, 0 \leq t \leq 2 \pi\) and \(a>0,\) with density \(\rho(x, y)=y\)
Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(C_{2}=-C_{1},\) then \(\int_{C_{1}} f(x, y) d s+\int_{C_{2}} f(x, y) d s=0\).
Find the total mass of two turns of a spring with density \(\rho\) in the shape of the circular helix \(\mathbf{r}(t)=3 \cos t \mathbf{i}+3 \sin t \mathbf{j}+2 t \mathbf{k}\) \(\rho(x, y, z)=z\)
Find the work done by the force field \(\mathbf{F}\) on a particle moving along the given path. \(\mathbf{F}(x, y, z)=y z \mathbf{i}+x z \mathbf{j}+x y \mathbf{k}\) \(C:\) line from (0,0,0) to (5,3,2)
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