The curves intersect when 6sin(5x)=6cos(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=6cos(5x), g(x)=y=6sin(5x).
Area A=6∫(cos(5x)-sin(5x))dx=(6/5)(sin(5x)+cos(5x))[0,π/20].
A=(6/5)(√2/2+√2/2-1)=(6/5)(√2-1).
1/A=⅚(√2+1).
Now we need x̄= (integrating by parts):
[6∫x(cos(5x)-sin(5x))dx]/A=
(6/5A)(xsin(5x)+⅕cos(5x)+xcos(5x)-⅕sin(5x))[0,π/20]=
(6/5A)(π√2/40+√2/10+π√2/40-√2/10-1/5)=
(6/5A)(π√2/20-⅕)=(6/25A)(π√2/4-1).
x̄=((√2+1)/5)(π√2/4-1).
ȳ=(36/2A)∫((cos²(5x)-sin²(5x))dx,
ȳ=(18/A)∫cos(10x)dx=(9/5A)(sin(10x)[0,π/20]=9/5A.
1/A=⅚(√2+1), so ȳ=3(√2+1)/2.
The centroid is (((√2+1)/5)(π√2/4-1),3(√2+1)/2).