What is the area of the trapezoid shown? Express your answer in simplest exact form.
[asy]
unitsize(1.5cm);
pair a=(0,0); pair b=(1,sqrt(3)); pair c=(3,sqrt(3)); pair d=(4,0);
draw(a--b--c--d--a);
label("2",(a+b)/2,NW);
label("2",(b+c)/2,N);
label("2",(c+d)/2,NE);
label("~$60^\circ$",a, NE);
label("$60^\circ$~",d, NW);
[/asy]