Above is the graph with x measured in radians. The area is the hump between x=0 and 1.57 ((pi)/2).
This area can be broken down into thin rectangles with height y and width dx so the integral is S[0,(pi)/2](ydx) where S is the integral sign and [ ] enclose the elements. Put y=2sin(2x):
S[0,(pi)/2](2sin(2x)dx)=-cos(2x)[0,(pi)/2]=-(0-1)=1.