Suppose $C$ is the line segment from the point $(0,0)$ to the point $(3,9)$ and $\vec{F}=5xy\,\vec i + 3x\,\vec j$.

(a) Is $\int_C \vec{F} \cdot d\vec{r}$ greater than, less than, or equal to zero? ? greater than zero less than zero equal to zero
(Be sure you are able to give a geometric explanation for your answer.)

(b) A parameterization of $C$ is $(x(t),y(t)) = (t,3 t)$ for $0 \leq t \leq 3$. Use this to compute $\int_C \vec{F} \cdot d\vec{r}$.
$\int_C \vec{F} \cdot d\vec{r} =$

(c) Suppose a particle leaves the point $(0,0)$, moves along the line toward the point $(3,9)$, stops before reaching it and backs up, stops again and reverses direction, then completes its journey to the endpoint. All travel takes place along the line segment joining the point $(0,0)$ to the point $(3,9)$. Call this path $C'$. How is $\int_{C'} \vec{F} \cdot d\vec{r}$ related to $\int_{C} \vec{F} \cdot d\vec{r}$?
$\int_{C'} \vec{F} \cdot d\vec{r}$ ? is greater than is less than is equal to $\int_{C} \vec{F} \cdot d\vec{r}$
(Be sure you can explain why this is.)

(d) A parameterization for a path like $C'$ is given by $$(x(t),y(t)) = \left(\frac{1}{27}\left(16\,t^3 - 72\,t^2 + 99\,t\right), \frac{1}{9}\left(16\,t^3 - 72\,t^2 + 99\,t\right))\right).$$ For what value of $t$ is this parameterization
at $(0,0)$? $t =$
at $(3,9)$? $t =$

What equation must $(x(t),y(t))$ satisfy?

(Enter your equation in terms of $x$ and $y$: thus, if $y(t) = 3 x(t)^2 - 6$, you would enter $y = 3x^2 - 6$.).
By hand, on a separate sheet of paper, verify that this equation is in fact satisfied by this parameterization.

At what values of $t$ does the particle change direction?
At $t =$ and

(e) Set up and find $\int_{C'} \vec{F} \cdot d\vec{r}$ using the parameterization in part (d).
With $a =$ and $b =$ ,
$\int_{C'} \vec{F} \cdot d\vec{r} = \int_a^b$ $dt =$
(Note whether you get the same answer as you did in part (b).)