EXAMPLE $3$ A wire takes the shape of a semicircle

x$2+$ y$2=49,$

y $0,$

and is thicker near its base than near the top. Find the center of mass of the wire if the linear density at any point is proportional to its distance from the line

y $=8.$

SOLUTION As in this example we use the parametrization

x $=7$ cos$($t$),$

y $=7$ sin$($t$),$

$0$ t $,$

and find that

ds $=7$ dt$.$

The linear density is

$($x$,$ y$)=$ k$(8$ y$)$