Question

Find the volume of the solid obtained by rotating about x-axis the region bounded by y = cosx and y = 0 and x = 0 and x = pi/2

Answer 1

\[\int\limits_{0}^{\frac{ \pi }{ 2 }} \pi \cos^2 x dx\]

Answer 2

∫cos²(x) dx= ∫[1-sin²(x)] dx=

=x- ∫sen²(x) dx

∫u dv=u.v-∫vdu

u=sin(x) and dv=sin(x) dx,

du=cos(x)

v=∫sin(x) dx=-cos(x)

∫sin²(x) dx= sin(x) cos(x)+∫cos²(x) dx

∫cos²(x) dx=x-∫sin²(x) dx=

=x-sin(x) cos(x)-∫cos²(x) dx

2.∫cos²(x) dx=x-sin(x) cos(x)

∫cos²(x).dx=[x-sin(x).cos(x)]/2

sin(x).cos(x)=sin(2x)/2

∫cos²(x).dx=x/2-sin(2x)/4

∫cos²(x).dx=[x-sin(x) cos(x)]/2

[pi/2-sin(pi/2) cos(pi/2)]/2 - [ 0 -sin 0 cos 0]/2 =
pi/4 - 0 = pi/4