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]