The curves intersect when 2sin(5x)=2cos(5x).
Therefore, tan(5x)=1, 5x=π/4, x=π/20.
First calculate the area between the curves for x∈[0,π/20].
f(x)=y=2cos(5x), g(x)=y=2sin(5x).
Area A=2∫(cos(5x)-sin(5x))dx=(2/5)(sin(5x)+cos(5x))[0,π/20].
A=⅖(√2/2+√2/2-1)=⅖(√2-1).
1/A=(5/2)(√2+1).
Now we need x̄= (integrating by parts):
[2∫x(cos(5x)-sin(5x))dx]/A=
(2/5A)(xsin(5x)+⅕cos(5x)+xcos(5x)-⅕sin(5x))[0,π/20]=
(2/5A)(π√2/40+√2/10+π√2/40-√2/10-1/5)=
(2/5A)(π√2/20-⅕)=(2/25A)(π√2/4-1).
x̄=((√2+1)/5)(π√2/4-1).
ȳ=(4/2A)∫((cos²(5x)-sin²(5x))dx,
ȳ=(2/A)∫cos(10x)dx=(1/5A)(sin(10x)[0,π/20]=1/5A.
1/A=(5/2)(√2+1), so ȳ=(√2+1)/2.
The centroid is (((√2+1)/5)(π√2/4-1),(√2+1)/2).