1. True or False A line integral $\int_{C}f(x,y)\,ds$ is equal to a definite integral if $C$ is a smooth curve defined on $[a,b]$, and if the function $f$ is continuous on some region that contains the curve $C.$

2. If $C$ is a smooth curve defined by the parametric equations $x=x(t)$ and $y=y(t),$ $a\leq t\leq b,$ and $f$ is a function that is continuous on some region that contains the curve $C$, then the line integral of $f$ along $C$ from $t=a$ to $t=b$ exists and is given by $\int_{C}f(x,y)\,ds=\int_{a}^{b}$__________$dt$.

3. True or False The value $\int_{C}f(x,y)\,ds$ depends on the choice of the parametric equations used to represent the smooth curve $C.$

4. If the variable mass density of a wire at a point $(x,y)$ is $\rho (x,y)$, then the mass $M$ of the wire can be found using the line integral _______, where $C$ is the curve representing the shape of the wire.

5. True or False $\int_{-C}(P\,dx+Q\,dy)= \int_{C}(P\,dx-Q\,dy)$

6. True or False A piecewise-smooth curve $C$ consists of a finite number of smooth curves that are joined together end to end.

7. If a piecewise-smooth curve $C$ consists of smooth curves $C_{1},\,C_{2},\,\ldots ,\,C_{n},$ $n\geq 2,$ then $\int_{C}(P\,dx+Q\,dy)=$_________.

8. True or False Suppose $C$ is a smooth curve defined by the parametric equations $x=x(t),$ $y=y(t),$ and $z=z(t),$ $\ a\leq t\leq b$, and $P(x,y,z),$ $Q(x,y,z),$ and $R(x,y,z)$ are functions that are continuous on some solid in space containing $C$, then $\int_{C}P(x,y,z)\,dx=\int_{a}^{b}P( x(t) ,y(t) ,z(t) ) x(t) \,dt.$

9. $\int_{C}x^{2}y\,ds;\quad C$: $x=\cos t$, $y=\cos t$ , $0\leq t\leq \dfrac{\pi }{2}$

10. $\int_{C}(y-x^{2})\, ds;\quad C$: $x=t$, $y=3t$, $0\leq t\leq 1$

11. $\int_{C}xy\,ds;\quad C$: $x=t^{2},$ $y=4t,$ $0\leq t\leq 2$

12. $\int_{C}x\,ds;\quad C$: $x=t$, $y=3t^{2}$, $0\leq t\leq 1$

13. $\int_{C}\dfrac{y}{2x^{2}-y^{2}}ds;\quad C$: $x=t$, $y=t$, $1\leq t\leq 5$

14. $\int_{C}\dfrac{x^{2}}{x^{3}+y^{3}}ds;\quad C$: $x=3t$, $y=4t$, $1\leq t\leq 2$

15. Find $\int_{C}2y\,ds$, where $C$ is the curve $y=\dfrac{1}{2}x^{3}$ joining $(0,0)$ to $(2,4)$.

16. Find $\int_{C}x\,ds$, where $C$ is the curve $x=5y^{3}$ joining $(0,0)$ to $(5,1)$.

17. Find $\int_{C}( xy+1) \,ds$, where $C$ is the curve $\mathbf{r}(t)=\sin t\mathbf{i}+\cos t\mathbf{j},$ $0\leq t\leq \pi$.

18. Find $\int_{C}xy^{2}\,ds$, where $C$ is the curve $\mathbf{r}(t)=\cos t \mathbf{i}+\sin t\mathbf{j},$ $0\leq t\leq \dfrac{\pi }{2}$.

19. $\int_{C}(x+y^{2})\,ds$

    1. (a) $C_{1}$: $\ x=t$, $y=t$, $0\leq t\leq 1$
    2. (b) $C_{2}:$ $\ x=\sin t$, $y=\sin t$, $0\leq t\leq \dfrac{\pi }{2}$

20. $\int_{C}(x+y)\,ds$

    1. (a) $C_{1}$: $\ x=t$, $y=3t$, $0\leq t\leq 1$
    2. (b) $C_{2}$: $\ x=\sin (2t)$, $y=3\sin (2t) ,$ $0\leq t\leq \dfrac{\pi }{2}$

21. $\int_{C}( x+e^{y}) \,ds$

    1. (a) $C_{1}$: $\ x=1-t$, $y=t$, $0\leq t\leq 1$
    2. (b) $C_{2}$: $\ x=\cos t$, $y=1-\cos t$, $0\leq t\leq \dfrac{\pi }{2}$

22. $\int_{C}xe^{x^{2}}ds$

    1. (a) $C_{1}$: $\ x=t+1$, $y=t$, $0\leq t\leq 1$
    2. (b) $C_{2}$: $\ x=1+\sin t$, $y=\sin t$, $0\leq t\leq \dfrac{\pi }{2}$

23. $f(x,y) =2x+y^{2}$ along the line $y=3-x$ from $\left( 0,3\right)$ to $(3,0) .$

24. $f(x,y) =xe^{y}$ along the line $2x-y=1$ from $(0,-1)$ to $(1,1) .$

25. $f(x,y) =xe^{y}$ along the curve $C$ traced out by $x=e^{2y}$ from $( 1,0)$ to $(e^{2},1) .$

26. $f(x,y) =x^{2} y$ along the parabola $x=2y^{2}$ from $(0,0)$ to $( 2\pi ,\sqrt{\pi }) .$

27. Find $\int_{C}(x^{2}y\,dx+xy\,dy)$ along the curve $C$: $x^{2}+y^{2}=1$ from $(1,0)$ to $(0,1)$ using the following parametric equations:

    1. (a) $x=\cos t$, $y=\sin t$, $0\leq t\leq \dfrac{\pi }{2}$
    2. (b) $x=\sqrt{1-y^{2}}$, $0\leq y\leq 1$

28. Find $\int_{C}(xy\,dx+xy^{2}\,dy)$ along the curve $C$: $x^{2}+y^{2}=9$ from $(3,0)$ to $(-3,0)$ traced out by the parametric equations:

    1. (a) $x=3\cos t$, $y=3\sin t$, $0\leq t\leq \pi$
    2. (b) $y=\sqrt{9-x^{2}}$, $-3\leq x\leq 3$

29. Find $\int_{C}[y^{2}\,dx+(xy-x^{2})\,dy]$ along each of the given curves from $(0,0)$ to $(1,3)$:

    1. (a) $C$ is the line $y=3x$.
    2. (b) $C$ is the parabola $y^{2}=9x$.

30. Find $\int_{C}[(x^{2}+y^{2})\,dx+3x^{2}y\,dy]$ along each of the given curves from $(-2,4)$ to $(2,4)$:

    1. (a) $C$ is the parabola $y=x^{2}$.
    2. (b) $C$ is the line $y=4$.

31. $C$ is the curve $y=x^{2}$ from $(0,0)$ to $(1,1)$.

32. $C$ is the curve $y=x^{3}$ from $(0,0)$ to $(1,1)$.

33. $C$ is the curve $x=\cos t$, $y=\sin t$, $0\leq t\leq \dfrac{\pi }{2}$.

34. $C$ is the curve $x=1-t$, $y=t$, $0\leq t\leq 1$.

39. Find $\int_{C}[(x\cos y)\,dx-(y\sin x)\,dy]$, where $C$ consists of line segments connecting the points $(0,0)$, $(1,0)$, $(1,1)$ , and $(0,1)$, in that order.

40. Find $\int_{C}[(x\sin y)\,dx-(y\cos x)\,dy]$, where $C$ consists of line segments connecting the points $(0,0)$, $(0,1)$, $(1,1)$, and $(1,0)$, in that order.

41. Find $\int_{C}(yz\,dx+xz\,dy+xy\,dz)$, where $C$ consists of line segments connecting the points $(0,0,0)$, $(1,0,0)$, $(1,1,0)$, and $(1,1,1)$, in that order.

42. Find $\int_{C}[yz\,dx+(y+zx)\,dy+xz\,dz]$, where $C$ consists of line segments connecting the points $(0,0,0)$, $(1,0,0)$, $(1,2,0)$, and $(1,2,1)$, in that order.

43. The line segment joining the two points

44. The parabola $y=4-x^{2}$

45. The line segment from $(2,0)$ to $(2,2)$, followed by the line segment from $(2,2)$ to $(0,4)$.

46. The parabola $y=4-x^{2}$ from $( 2,0)$ to $( 1,3) ,$ followed by the line segment from $( 1,3)$ to $( 0,4)$.

47. $\mathbf{F}(x,y)=(x+2y)\mathbf{i}+(2x+y)\mathbf{j}$;
    $C$ is the curve $\mathbf{r}(t)=t\mathbf{i}+t^{2}\mathbf{j}$, $0\leq t\leq 1$.

48. $\mathbf{F}(x,y)=x^{2}\mathbf{i}+xy\mathbf{j}$
    $C$ is the curve $\mathbf{r}(t)=(\cos t)\mathbf{i}+(\sin t)\mathbf{j}$, $0\leq t\leq \pi$.

49. $\mathbf{F}(x,y,z)=xy\mathbf{i}+x^{2}z\mathbf{j}+xyz\mathbf{k}$
    $C$ is the curve $\mathbf{r}(t)=e^{t}\mathbf{i}+e^{-t}\mathbf{j}+t^{2} \mathbf{k}$, $0\leq t\leq 1$.

50. $\mathbf{F}(x,y,z)=y\mathbf{i}+x^{2}y\mathbf{j}+xz\mathbf{k}$
    $C$ is the curve $\mathbf{r}(t)=e^{2t}\mathbf{i}+e^{t}\mathbf{j}+t\mathbf{k}$, $0\leq t\leq 1$.

51. $C$ is the line segment from $(0,0,0)$ to $(1,0,0)$.

52. $C$ is the line segment from $(1,0,0)$ to $(1,2,0)$.

53. $C$ is the line segment from $(1,2,0)$ to $(1,2,3)$.

54. $C$ is the line segment from $(0,0,0)$ to $(1,2,3)$.

55. $z=f(x,y)=1-x^{2};\quad C$: $x=\sin t$, $y=\cos t$, $0\leq t\leq 2\pi$

56. $z=f(x,y)=1-y^{2};\quad C$: $x=\sin t$, $y=\cos t$, $0\leq t\leq 2\pi$

57. $z=f(x,y)=2xy;\quad C$: $y=\sqrt{1-x^{2}}$, $0\leq x\leq 1$

58. $z=f(x,y)=x+y;\quad C$: $y=\sqrt{1-x^{2}}$, $0\leq x\leq 1$

59. Find $\int_{C}\dfrac{\,ds}{x^{2}+y^{2}+z^{2}}$, where $C$ is the helix $x=a\cos t$, $y=a\sin t$, $z=bt$, $0\leq t\leq 1$.

60. Find $\int_{C}(y^{n}\,dx+x^{n}\,dy)$, where $C$ is the ellipse $x=a\sin t$, $y=b\cos t$, $0\leq t\leq 2\pi$, and $n \ge 1$ is an integer.

61. Find $\int_{C}y^{2}\,ds$, where $C$ is the first arch of the cycloid $x=a(t-\sin t)$, $y=a(1-\cos t)$.

62. Find $\int_{C}\sqrt{x^{2}+y^{2}}\,ds$, where $C$ is the curve $x=a(\cos t+t\sin t)$, $y=a(\sin t-t\cos t)$, $0\leq t\leq 2\pi$, $a > 0$.

63. Find $\int_{C}(z\,dx+x\,dy+y\,dz)$, where $C$ is the circular helix $x=a\cos t$, $y=a\sin t$, $z=t$, $0\leq t\leq 2\pi$.

64. Find $\int_{C}[xz\,dx+(y+z)\,dy+x\,dz]$, where $C$ is the curve $x=e^{t}$, $y=e^{-t}$, $z=e^{2t}$ from $t=0$ to $t=1$.

65. $C$: $y=x^{2}$, $1\leq x\leq 2;\quad \rho =\rho (x,y)=4x$

66. $C$: $x=3\sin t$, $y=3\cos t$, $0\leq t\leq \dfrac{\pi }{2};\quad \rho =\rho (x,y)=x+y+1$

67. Mass of a Wire Find the mass of a thin wire in the shape of a parabola $x=2t^{2},$ $y=t$, $-2\leq t\leq 3$, if the variable mass density of the wire is $\rho (x,y) =3x.$

68. Mass of a Wire Find the mass of a thin wire in the shape of a parabola $x=t,$ $y=t^{2}-1,$ $0\leq t\leq 3,$ if the variable mass density of the wire is $\rho (x,y) =x\left( y+1\right) .$

69. Mass of a Wire A spring is made of a thin wire twisted into the shape of a circular helix $x=2\cos t$, $y=2\sin t,$ $z=t.$ Find the mass of two turns of the spring if the wire has constant mass density $\rho$.

989

70. Mass of a Wire A spring is made of a thin wire twisted into the shape of a circular helix $x=3\sin t$, $y=3\cos t,$ $z=t.$ Find the mass of two turns of the spring if the wire has variable mass density $\rho ( x,y,z) =z+2.$

71. Mass of a Wire in Space Write a formula similar to $M=\int_{C}\rho (x,y)\, ds$ for the mass of a thin wire in the shape of a smooth space curve $C$ whose variable mass density is $\rho =\rho (x,y,z)$.

72. Mass of a Wire in Space The thin wire is in the shape of the helix $C$: $x=2\cos t$, $y=2\sin t$, $z=5t$, $0\leq t\leq 2\pi$, whose mass density is $\rho (x,y,z)=k$, a constant.

73. Mass of a Wire in Space The thin wire is in the shape of a helix $C$: $x=3\cos t$, $y=3\sin t$, $z=4t$, $0\leq t\leq 2\pi$, whose variable mass density is $\rho (x,y,z)=2+z$.

74. Mass of a Wire in Space The thin wire is in the shape of a helix $C$: $x=2\cos t$, $y=2\sin t$, $z=5t$, $0\leq t\leq 2\pi$, if the mass density is proportional to the square of the distance from the origin.

75. Lateral Surface Area Derive the formula $A=2\pi Rh$ for the surface area of a right circular cylinder using the line integral $\int_{C}f(x,y)\,ds$, where $z=f(x,y)=h$, $h>0$, and $C$ is the circle $x^{2}+y^{2}=R^{2}$.

76. Let $C_{1}$ be the curve $x=\cos \theta$, $y=\sin \theta$, $0\leq \theta \leq 4\pi$, and let $C_{2}$ be the curve $x=\cos t^{2}$, $y=\sin t^{2}$, $0\leq t\leq 2\sqrt{\pi }$. For $P(x,y) =x^{3}y$ and $Q(x,y) =y^{2}x$, show that $\int_{C_{1}}(P\,dx+Q\,dy)=\int_{C_{2}}(P\,dx+Q\,dy)$. Explain why this is so.

77. Find $\int_{C}(x+y)\,ds$, where $C$ is the curve $x=t$, $y=\dfrac{3t^{2}}{\sqrt{2}}$, $0\leq t\leq 1$.

78. Center of Mass Find the center of mass of a homogeneous wire in the shape of the hypocycloid

    1. (a) $x=a\cos ^{3}t,y=a\sin ^{3}t,\ 0\leq t\leq \dfrac{\pi }{2}$
    2. (b) $x=a\cos ^{3}t,y=a\sin ^{3}t,\ 0\leq t\leq 2\pi$

79. Suppose the line integral of $f(x,y)$ along the curve $C$ exists.

    1. (a) Use the Mean Value Theorem to derive the formula $$\int_{C}f(x,y)\,ds=\int_{a}^{b}f(x(t),y(t))\sqrt{\left( \dfrac{dx}{dt} \right) ^{\!\!2}+\left( \dfrac{dy}{dt}\right) ^{\!\!2}}\,dt$$
    2. (b) Apply the formula in (a) to the quantity $S(t)=\int_{a}^{t}\sqrt{(x')^2+(y')^2}\,du$, where $S' (t_{0})$ is the length of the portion of the curve traced out as $t$ moves from $a$ to $t_{0}$.

80. It follows from Problem 79 that the value of a line integral of a function along a smooth curve, given the hypotheses of the theorem, is independent of the parameterization. Why? In particular, if $x$ and $y$ are differentiable functions of the arc length as one moves along the curve, describe what form the integral $$\int_{C}f(x,y)\,ds=\int_{a}^{b}f(x(t),y(t)) \sqrt{\left( \dfrac{dx}{dt}\right) ^{\!\!2}+\left( \dfrac{dy}{dt}\right) ^{\!\!2}} \,dt$$

    takes when $s$ is used as the parameter.

81. We have seen two kinds of line integrals: $\int_{C}( P\,dx+Q\,dy)$ and $\int_{C}f\,ds$. These are closely connected.

    1. (a) If $C$ is expressed using arc length as the parameter, show that $\int_{c}(Pdx + Qdy)$ can be written in the form $\int_{c} fds$.
    2. (b) If the arc length $s$ along $C$ can be expressed as a function of $x$ and $y$, show that $\int_{c}fds$ can be written in the form $\int_{c}(Pdx + Qdy)$. For example, $$\int_{C}(x+y^{2})\,ds=\int_{C}\left[\dfrac{x^{2}+xy^{2}}{\sqrt{x^{2}+y^{2}}}\,dx+ \dfrac{xy+y^{3}}{\sqrt{x^{2}+y^{2}}}\,dy\right]$$ where $C$ is the curve $x=t$, $y=t$, for $0\leq t\leq 1$.