$1(5pts).$ Write down the integral for arclength $of$ the parabola

$(x,y)=(1-t^2,t),$tin$(-1,1)$

$2(15pts).$ Let $(t)=(e^{-t}cost,e^{-t}sint),$tinR.

$(5pts).$ Explain why $(t)is$ a parametrized smooth curve.

Hint: $It$ suffices $to$ justify that each componen$\frac{t}{e}ntryis$ a smooth function. $In$ calculus,

$we$ learned that product $of$ smooth functions $is$ also smooth.

$(5pts).$ Show that $ast,(t)$ approaches the origin $0$ and spins around $it.$

Hint: Draw a picture $of$ the curve and say something about the infinite spinnin$\frac{g}{r}$otation

$of$ the curve with shrinking radius.

$(5pts).$ Find the length $of$ the curve $i.e.$ the value $of$ the improper integral $\underset{t}{lim}{\displaystyle \underset{t_0}{\overset{t}{}}}|{}^{\text{'}}(t)|dt.$

$Isit$ finite $or?$

Hint: The arc$-$length should depend $on{t}_0.$