$3.$ Consider the curve $($denoted $\backslash ($ C$\_\{1\}\backslash ))$ whose equation is $\backslash ($ y$=2$ x$^\{3\}\backslash )$ where $\backslash (0\backslash$leq x $\backslash$leq $1\backslash )$ and answer the following questions for finding the center of mass of a thin wire lying along $\backslash ($ C$\_\{1\}\backslash )$ whose line density function at point $\backslash (($x$,$ y$)\backslash )$ is $\backslash (\backslash$delta$=$x $\backslash ).$

$($a$)$ Sketch the curve $\backslash ($ C$\_\{1\}\backslash )$ on the $\backslash ($ x y $\backslash )-$plane. Clearly label the axes, scales, cartesian coordinates of the end points and then draw a typical "sub$-$arc".

$($b$)$ Write out bullet by bullet $($in terms of $\backslash ($ x $\backslash ))$ the arc length, center of mass, mass of the typical "sub$-$arc" you draw and interval of integration.

$($c$)$ Write out bullet by bullet $($in terms of $\backslash ($ x $\backslash ))$ definite integrals for the total mass, total moment about the $\backslash ($ y $\backslash )-$axis, total moment about the $\backslash ($ x $\backslash )-$axis of the thin wire, respectively, and then write out the coordinates for the center of mass.

Resource: Sce supplemental video under "Tasks Week $4-$ Tuesday $(01/28)"$ from $16$th min onwards.

$($d$)$ Rewrite the curve $\backslash ($ C$\_\{1\}\backslash )$ as an equation of $\backslash ($ x $\backslash )$ in terms of $\backslash ($ y $\backslash )$ and indicate the corresponding interval, then rewrite the line density function as a function in terms of $\backslash ($ y $\backslash ).$

$($e$)$ Write out bullet by bullet $($in terms of $\backslash ($ y $\backslash ))$ the are length, center of mass, mass of the typical "sub$-$arc" you draw and interval of integration.

$($f$)$ Write out bullet by bullet $($in terms of $\backslash ($ y $\backslash ))$ definite integrals for the total mass, total moment about the $\backslash ($ y $\backslash )-$axis, total moment about the $\backslash ($ x $\backslash )-$axis of the thin wire, respectively, and then write out the coordinates for the center of mass.

Resource: See supplemental video under "Tasks Week $4-$ Tuesday $(01/28)"$ from $33$:$30$th min onwards.