# Amath

Revision Exercise 20

Q16

In Fig. 14, the shaded region is bounded by the curves y=cos x,y=sin 2x, and the x-axis within the range (x is equal or bigger that 0 and

pai/2 is smaller or equal than x)

(a) Find the coordinates of P

(b) If the region is rotated about the x-axis ,find the volume of the solid generated in term of pai.

Rating

(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)

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)