Section 1
Line Integrals
In each of Problems 1 through 15, evaluate the line integral.$\int_C x d x-d y+z d z$, with $C$ given by $x(t)=t, y(t)=$ $t, z(t)=t^3$ for $0 \leq t \leq 1$
In each of Problems 1 through 15, evaluate the line integral.$\int_C-4 x d x+y^2 d y-y z d z$, with $C$ given by $x(t)=$ $-t^2, y(t)=0, z(t)=-3 t$ for $0 \leq t \leq 1$
In each of Problems 1 through 15, evaluate the line integral.$\int_C(x+y) d s$, where $C$ is given by $x=y=t, z=t^2$ for $0 \leq t \leq 2$
In each of Problems 1 through 15, evaluate the line integral.$\int_C x^2 z d s$, where $C$ is the line segment from $(0,1,1)$ to $(1,2,-1)$
In each of Problems 1 through 15, evaluate the line integral.$\int_C \mathbf{F} \cdot d \mathbf{R}$, where $\mathbf{F}=\cos (x) \mathbf{i}-y \mathbf{j}+x z \mathbf{k}$ and $\mathbf{R}=$ $t \mathbf{i}-t^2 \mathbf{j}+\mathbf{k}$ for $0 \leq t \leq 3$
In each of Problems 1 through 15, evaluate the line integral.$\int_c 4 x y d s$, with $C$ given by $x=y=t, z=2 t$ for $1 \leq t \leq 2$
In each of Problems 1 through 15, evaluate the line integral.$\int_C \mathbf{F} \cdot d \mathbf{R}$, with $\mathbf{F}=x \mathbf{i}+y \mathbf{j}-z \mathbf{k}$ and $C$ the circle $x^2+y^2=4, z=0$, going around once counterclockwise.
In each of Problems 1 through 15, evaluate the line integral.$\int_C y z d s$, with $C$ the parabola $z=y^2, x=1$ for $0 \leq$ $y \leq 2$
In each of Problems 1 through 15, evaluate the line integral.$\int_C-x y z d z$, with $C$ the curve $x=1, y=\sqrt{z}$ for $4 \leq$ $z \leq 9$
In each of Problems 1 through 15, evaluate the line integral.$\int_C x z d y$, with $C$ the curve $x=y=t, z=-4 t^2$ for $1 \leq t \leq 3$
In each of Problems 1 through 15, evaluate the line integral. $\int_C 8 z^2 d s$, with $C$ the curve $x=y=2 t^2, z=1$ for $1 \leq t \leq 2$
In each of Problems 1 through 15, evaluate the line integral.$\int_c \mathbf{F} \cdot d \mathbf{R}$, with $\mathbf{F}=\mathbf{i}-x \mathbf{j}+\mathbf{k}$ and $\mathbf{R}=\cos (t) \mathbf{i}-$ $\sin (t) \mathbf{j}+t \mathbf{k}$ for $0 \leq t \leq \pi$
In each of Problems 1 through 15, evaluate the line integral. $\int_C 8 x^2 d y$, with $C$ given by $x=e^t, y=-t^2, z=t$ for $1 \leq t \leq 2$
In each of Problems 1 through 15, evaluate the line integral.$\int_C x d y-y d x, C$ the curve $x=y=2 t, z=e^{-t}$ for $0 \leq t \leq 3$
In each of Problems 1 through 15, evaluate the line integral.$\int_C \sin (x) d s$, with $C$ given by $x=t, y=2 t, z=3 t$ for $1 \leq t \leq 3$
Find the mass and center of mass of a thin, straight wire from the origin to $(3,3,3)$ if $\delta(x, y, z)=x+y+$ $z$ grams per centimeter.
Find the work done by $\mathbf{F}=x^2 \mathbf{i}-2 y z \mathbf{j}+z \mathbf{k}$ in moving an object along the line segment from $(1,1,1)$ to $(4,4,4)$.
Show that any Riemann integral $\int_a^b f(x) d x$ is equal to a line integral $\int_c \mathbf{F} \cdot d \mathbf{R}$ for appropriate choices of $\mathbf{F}$ and $C$. In this sense the line integral generalizes the Riemann integral.