Amath

(a)
cos x =sin 2x
cos x =2sinxcosx
cosx(1-2sinx)=0
cosx=0 or sinx=1/2
x= π/2 or π/6
y= 0 or √3/2
P(π/6,√3/2)
(b)

圖片參考：http://www.hivepc.com/calculus/survivalguide/img/disk_x.gif

The volume of the solid generated
=π∫(cos^2 x-sin^2 2x) dx from [0 to π/6]
=π∫(cos^2 x- 4sin^2 xcos^2 x) dx
= π∫(4cos^4 x- 3cos^2 x) dx
Let In=∫(cos^n x) dx
∫(cos^2 x) dx
=(sin x cos x)/2+(1/2) I0 [x from 0 to π/6]
=(1/2)(√3/2)/2+(1/2)(π/6)
=(√3/8)+(π/12)
∫(cos^4 x) dx
=(sin x cos^3 x)/4+(3/4) I2 [x from 0 to π/6]
=(1/2)(3√3/4)(1/4)+(3/4)[(√3/8)+(π/12)]
=(3√3/16)+(π/16)
π∫(4cos^4 x- 3cos^2 x) dx
=π[(3√3/4)+(π/4)-(3√3/8)-(π/4)]
=π(3√3/8)