The coordinates of the center of mass of a wire with variable density are given by x My

In Problems 1–4, evaluate \(\int_{C} G(x, y) d x, \int_{C} G(x, y) d y\) and \(\int_{C} G(x, y) d s\) on the indicated curve C. \(G(x, y)=2 x y ; x=5 \cos t, y=5 \sin t, 0 \leq t \leq \pi / 4\)

In Problems 1–4, evaluate \(\int_{C} G(x, y) d x, \int_{C} G(x, y) d y\) and \(\int_{C} G(x, y) d s\) on the indicated curve C. \(G(x, y)=x^{3}+2 x y^{2}+2 x ; x=2 t, y=t^{2}, 0 \leq t \leq 1\)

In Problems 1–4, evaluate \(\int_{C} G(x, y) d x, \int_{C} G(x, y) d y\) and \(\int_{C} G(x, y) d s\) on the indicated curve C. \(G(x, y)=3 x^{2}+6 y^{2} ; y=2 x+1,-1 \leq x \leq 0\)

In Problems 1–4, evaluate \(\int_{C} G(x, y) d x, \int_{C} G(x, y) d y\) and \(\int_{C} G(x, y) d s\) on the indicated curve C. \(G(x, y)=x^{2} / y^{3} ; 2 y=3 x^{2 / 3}, 1 \leq x \leq 8\)

In Problems 5 and 6, evaluate \(\int_{C} G(x, y, z) d x, \int_{C} G(x, y, z) d y\), \(\int_{C} G(x, y, z) d z, \text { and } \int_{C} G(x, y, z) d s\) ds on the indicated curve C \(G(x, y, z)=z ; x=\cos t, y=\sin t, z=t, 0 \leq t \leq \pi / 2\)

In Problems 5 and 6, evaluate \(\int_{C} G(x, y, z) d x, \int_{C} G(x, y, z) d y\), \(\int_{C} G(x, y, z) d z, \text { and } \int_{C} G(x, y, z) d s\) ds on the indicated curve C \(G(x, y, z)=4 x y z ; x=\frac{1}{3} t^{3}, y=t^{2}, z=2 t, 0 \leq t \leq 1\)

In Problems 7–10, evaluate \(\int_{C}(2 x+y) d x+x y d y\) on the given curve C between (1, 2) and (2, 5). y = x + 3

In Problems 7–10, evaluate \(\int_{C}(2 x+y) d x+x y d y\) on the given curve C between (1, 2) and (2, 5). \(y=x^{2}+1\)

In Problems 7–10, evaluate \(\int_{C}(2 x+y) d x+x y d y\) on the given curve C between (1, 2) and (2, 5).

In Problems 7–10, evaluate \(\int_{C}(2 x+y) d x+x y d y\) on the given curve C between (1, 2) and (2, 5).

In Problems 11–14, evaluate \(\int_{C} y d x+x d y\) on the given curve C between (0, 0) and (1, 1). \(y=x^{2}\)

In Problems 11–14, evaluate \(\int_{C} y d x+x d y\) on the given curve C between (0, 0) and (1, 1). y = x

In Problems 11–14, evaluate \(\int_{C} y d x+x d y\) on the given curve C between (0, 0) and (1, 1). C consists of the line segments from (0, 0) to (0, 1) and from (0, 1) to (1, 1).

In Problems 11–14, evaluate \(\int_{C} y d x+x d y\) on the given curve C between (0, 0) and (1, 1). C consists of the line segments from (0, 0) to (1, 0) and from (1, 0) to (1, 1).

Evaluate \(\int_{C}\left(6 x^{2}+2 y^{2}\right) d x+4 x y d y\), where C is given by \(x=\sqrt{t}, y=t, 4 \leq t \leq 9\)

Evaluate \(\int_{C}-y^{2} d x+x y d y\), where C is given by x = 2t, \(y=t^{3}, 0 \leq t \leq 2\)

Evaluate \(\int_{C} 2 x^{3} y d x+(3 x+y) d y\), where C is given by\(x=y^{2}\) from (1, 1) to (1, 1).

Evaluate \(\int_{C} 4 x d x+2 y d y\), where C is given by \(x=y^{3}+1\) from (0, 1) to (9, 2).

In Problems 19 and 20, evaluate \(\oint_{C}\left(x^{2}+y^{2}\right) d x-2 x y d y\) on the given closed curve C.

In Problems 19 and 20, evaluate \(\oint_{C}\left(x^{2}+y^{2}\right) d x-2 x y d y\) on the given closed curve C.

In Problems 21 and 22, evaluate \(\oint_{C} x^{2} y^{3} d x-x y^{2} d y\) on the given closed curve C.

In Problems 21 and 22, evaluate \(\oint_{C} x^{2} y^{3} d x-x y^{2} d y\) on the given closed curve C.

Evaluate \(\oint_{C}\left(x^{2}-y^{2}\right) d s\), where C is given by \(x=5 \cos t, \quad y=5 \sin t, \quad 0 \leq t \leq 2 \pi\)

Evaluate \(\int_{-C} y d x-x d y\), where C is given by \(x=2 \cos t, \quad y=3 \sin t, \quad 0 \leq t \leq \pi .\)

In Problems 25–28, evaluate \(\int_{C} y d x+z d y+x d z\) on the given curve C between (0, 0, 0) and (6, 8, 5). C consists of the line segments from (0, 0, 0) to (2, 3, 4) and from (2, 3, 4) to (6, 8, 5).

In Problems 25–28, evaluate \(\int_{C} y d x+z d y+x d z\) on the given curve C between (0, 0, 0) and (6, 8, 5). \(x=3 t, y=t^{3}, z=\frac{5}{4} t^{2}, \quad 0 \leq t \leq 2\)

In Problems 25–28, evaluate \(\int_{C} y d x+z d y+x d z\) on the given curve C between (0, 0, 0) and (6, 8, 5).

In Problems 25–28, evaluate \(\int_{C} y d x+z d y+x d z\) on the given curve C between (0, 0, 0) and (6, 8, 5).

In Problems 29 and 30, evaluate \(\int_{C} \mathbf{F} \cdot d \mathbf{r}\) \(\mathbf{F}(x, y)=y^{3} \mathbf{i}-x^{2} y \mathbf{j} ; \mathbf{r}(t)=e^{-2 t} \mathbf{i}+e^{2} \mathbf{j}, 0 \leq t \leq \ln 2\)

In Problems 29 and 30, evaluate \(\int_{C} \mathbf{F} \cdot d \mathbf{r}\) \(\mathbf{F}(x, y, z)=e^{x} \mathbf{i}+x e^{x y} \mathbf{j}+x y e^{x y z} \mathbf{k} ; \mathbf{r}(t)=t \mathbf{i}+t^{2} \mathbf{j}+t^{3} \mathbf{k}, 0 \leq t \leq 1 \)

Find the work done by the force \(\mathbf{F}(x, y)=y \mathbf{i}+x \mathbf{j}\) acting along y=ln x from (1, 0) to (e, 1).

Find the work done by the force \(\mathbf{F}(x, y)=2 x y \mathbf{i}+4 y^{2} \mathbf{j}\) acting along the piecewise-smooth curve consisting of the line segments from (2, 2) to (0, 0) and from (0, 0) to (2, 3).

Find the work done by the force \(\mathbf{F}(x, y)=(x+2 y) \mathbf{i}+ (6 y-2 x) \mathbf{j}\) acting counterclockwise once around the triangle with vertices (1, 1), (3, 1), and (3, 2).

Find the work done by the force \(\mathbf{F}(x, y, z)=y z \mathbf{i}+x z \mathbf{j}+x y \mathbf{k}\) acting along the curve given by \(\mathbf{r}(t)=t^{3} \mathbf{i}+t^{2} \mathbf{j}+t \mathbf{k}\) from t=1 to t=3.

Find the work done by a constant force \(\mathbf{F}(x, y)=a \mathbf{i}+b \mathbf{j}\) acting counterclockwise once around the circle \(x^{2}+y^{2}=9\)

In an inverse square force field \(\mathbf{F}=c \mathbf{r} /\|\mathbf{r}\|^{3}\), where c is a constant and \(\mathbf{r}=x \mathbf{i}+y \mathbf{j}+z \mathbf{k}\),* find the work done in moving a particle along the line from (1, 1, 1) to (3, 3, 3).

Verify that the line integral \(\int_{C} y^{2} d x+x y d y\) has the same value on C for each of the following parameterizations: \(C: x=2 t+1, \quad y=4 t+2, \quad 0 \leq t \leq 1\) \(C: x=t^{2}, y=2 t^{2}, 1 \leq t \leq \sqrt{3}\) \(C: x=\ln t , y=2 \ln t, e \leq t \leq e^{3}\)

Consider the three curves between (0, 0) and (2, 4): \(C_{1}: x=t, \quad y=2 t, \quad 0 \leq t \leq 2\) \(C_{2}: x=t, \quad y=t^{2}, \quad 0 \leq t \leq 2\) \(C_{3}: x=2 t-4, \quad y=4 t-8, \quad 2 \leq t \leq 3\) Show that \(\int_{C_{1}} x y d s=\int_{C_{3}} x y d s, \text { but } \int_{C_{1}} x y d s \neq \int_{C_{2}} x y d s\). Explain.

Assume a smooth curve C is described by the vector function r(t) for \(a \leq t \leq b\). Let acceleration, velocity, and speed be given by \(\mathbf{a}=d \mathbf{v} / d t, \mathbf{v}=d \mathbf{r} / d t, \text { and } v=\|\mathbf{v}\|\), respectively. Using Newton’s second law F=m a, show that, in the absence of friction, the work done by F in moving a particle of constant mass m from point A at t=a to point B at t=b is the same as the change in kinetic energy: \(K(B)-K(A)=\frac{1}{2} m[v(b)]^{2}-\frac{1}{2} m[v(a)]^{2} .\) [Hint: Consider \(\frac{d}{d t} v^{2}=\frac{d}{d t} \mathbf{v} \cdot \mathbf{v}\).]

If \(\rho(x, y)\) is the density of a wire (mass per unit length), then \(m=\int_{C} \rho(x, y) d s\) is the mass of the wire. Find the mass of a wire having the shape of the semicircle x=1+cos t, y=sin t, \(0 \leq t \leq \pi\), if the density at a point P is directly proportional to distance from the y-axis.

The coordinates of the center of mass of a wire with variable density are given by \(\bar{x}=M_{y} / m, \bar{y}=M_{x} / m\), where \(m=\int_{C} \rho(x, y) d s, \quad M_{x}=\int_{C} y \rho(x, y) d s\) and \(M_{y}=\int_{C} x \rho(x, y) d s\) Find the center of mass of the wire in Problem 40.

A force field F(x, y) acts at each point on the curve C, which is the union of \(C_{1}, C_{2}, \text { and } C_{3}\) shown in red in FIGURE 9.8.21. ||F|| is measured in pounds and distance is measured in feet using the scale given in the figure. Use the representative vectors shown to approximate the work done by F along C. [Hint: Use \(W=\int_{C} \mathbf{F} \cdot \mathbf{T} d s\).]