{{{id=1|
var('t')
polar_plot(4*sin(t),(t,0,pi))
///
}}}

{{{id=2|
polar_plot(4*sin(t),(t,0,pi), thickness=3)+polar_plot(2,(t,0,2*pi),thickness=3,color="red")
///
}}}

{{{id=3|
polar_plot(4*sin(t),(t,0,pi), thickness=3)+polar_plot(2,(t,0,2*pi),thickness=3,color="red")+line([(0,0),(1.5*sqrt(3),1.5)], thickness = 3, color="green")+line([(0,0),(-1.5*sqrt(3),1.5)], thickness = 3, color="green")
///
}}}

<p><span style="font-size: large;">The Mass</span></p>

{{{id=4|
var('r t');
integral(integral(r*cos(t)^2,(r,2,4*sin(t))),(t,pi/6,5*pi/6))
///
1/4*(3*sqrt(3))
}}}

{{{id=11|
M=integral(integral(r*cos(t)^2,(r,2,4*sin(t))),(t,pi/6,5*pi/6))
///
}}}

{{{id=12|
M
///
1/4*(3*sqrt(3))
}}}

<p><span style="font-size: x-large;">The $x$-</span><span style="font-size: x-large;">moment, $M_x$</span></p>

{{{id=5|
Mx=integral(integral(r^2*sin(t)*cos(t)^2,(r,2,4*sin(t))),(t,pi/6,5*pi/6))
Mx
///
8/9*pi + 1/18*(3*sqrt(3)) + 1/6*sqrt(3) - 1/3*sqrt(3)
}}}

{{{id=7|
Mx.simplify()
///
8/9*pi
}}}

{{{id=9|
y_bar = Mx/M
y_bar.simplify()
///
32/81*sqrt(3)*pi
}}}

{{{id=10|
y_bar.simplify().n()
///
2.14968813538870
}}}

<p><span style="font-size: x-large;">Check the $y$-moment, $M_y$</span></p>

{{{id=13|
My=integral(integral(r^2*cos(t)*cos(t)^2,(r,2,4*sin(t))),(t,pi/6,5*pi/6))
My
///
0
}}}

{{{id=15|

///
}}}