$($a$)$ Sketch the curve $y^3=x^2.$

$($b$)$ Use the following formulas to set up two integrals for the arc length from $(0,0)$ to $(1,1).$ Observe that one of these is an improper integral.

$($I$)L={\displaystyle \underset{a}{\overset{b}{}}}\sqrt[\phantom{2}]{1+(\frac{dy}{dx}{)}^2}dx$

$($II$)L={\displaystyle \underset{c}{\overset{d}{}}}\sqrt[\phantom{2}]{1+(\frac{dx}{dy}{)}^2}dy$

$L={\displaystyle \underset{0}{\overset{1}{}}}(x)dx=$

$$

$($c$)$ Find the length of the arc of this curve from $(-27,9)$ to $(8,4).$