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\flushleft\underline{Homework 10 --- MATH 1586 Spring 2018} \\
Let $C$ be a curve that is parameterized by $\vec{r}(t)$ for $a \leq t \leq b$. Recall that the $\mathrm{d}s$-type line integral is: 
$$\displaystyle\int_C f(x,y,z) \mathrm{d}s = \displaystyle\int_a^b f(x(t),y(t)) \lVert \vec{r}{\hspace{2pt}'}(t) \rVert \mathrm{d}t,$$
the $\mathrm{d}x$-type is
$$\displaystyle\int_C f(x,y,z) \mathrm{d}x = \displaystyle\int_a^b f(x(t),y(t)) x'(t) \mathrm{d}t,$$
the $\mathrm{d}y$-type is
$$\displaystyle\int_C f(x,y,z) \mathrm{d}y = \displaystyle\int_a^b f(x(t),y(t)) y'(t) \mathrm{d}t,$$
and the vector field line integral is
$$\displaystyle\int_C \vec{F} \cdot \mathrm{d}\vec{r} = \displaystyle\int_a^b \vec{F}(\vec{r}(t)) \cdot \vec{r}{\hspace{2pt}'}(t) \mathrm{d}t.$$
Also recall the ``tricks" to parametrize the line segment between two points: the line segment from the point $(a,b)$ to the point $(c,d)$ can be parametrized by $\vec{r}(t) = (1-t) \langle a,b \rangle + t \langle c,d \rangle$ for $0 \leq t \leq 1$. To parametrize a curve given by a function $y=h(t)$, use $\vec{r}(t) = \langle t, h(t) \rangle$ and choose $a \leq t \leq b$ appropriately.
\begin{enumerate}[1.]
\item Compute $\displaystyle\int_C x \mathrm{d}s$ where $C$ is the curve given by $\vec{r}(t)=\langle t^2, t \rangle 0 \leq t \leq 2$.
\item Set up \textbf{but do not evaluate} the following line integral: $\displaystyle\int_C (xy+\ln(x)) \mathrm{d}y$ where $C$ is the arc of the function $y=x^3$ from $(1,1)$ to $(2,8)$. 
\item Set up \textbf{but do not evaluate} the following line integral: $\displaystyle\int_C xe^{xy} \mathrm{d}x$, where $C$ is the arc of the curve $y=e^x$ from $(0,1)$ to $(2,e^2)$. 
\item Calculate $\displaystyle\int_C \vec{F} \mathrm{d} \vec{r}$, where $\vec{F}=\langle x^2y^3, -y\sqrt{x} \rangle$ and $\vec{r}(t)=\langle t^2, -t^3 \rangle$ and $0 \leq t \leq 1$. 
\item Set up \textbf{but do not evaluate} the following line integral: $\displaystyle\int_C \vec{F} \mathrm{d}\vec{r}$ where $\vec{F}=\langle \sin(x), \cos(y), xz \rangle$ and $C$ is the curve $\vec{r}(t)=\langle t^3, -t^2, t \rangle$ and $0 \leq t \leq 1$. 
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