Please check- Volume of solid y=2cosx

[confused_07]

This just seemed a little to easy, so I am sure there is something that I missed. Please check my work... Thanks.

R is bounded below by the x-axis and above by the curve y=2cosx, [0,(pi/2)]. Find the volume of the solid generated by revolving R around the y-axis by the method of cylindrical shells.

V= int [a,b] 2*pi*x*f[x]*dx
= int [0,(pi/2)] 2*pi*x(2cosx) dx
= 4*pi int [0,(pi/2)] x (cosx) dx
= 4*pi [sinx] [0,(pi/2)]
= 2sin*(pi^2)

 

[soroban]

Hello, confused_07!

This just seemed a little to easy, so I am sure there is something that I missed. . Yes!

R is bounded below by the x-axis and above by the curve \(\displaystyle y\:=\:2\cdot\cos x,\;\left[0,\,\frac{\pi}{2}\right]\)

Find the volume of the solid generated by revolving R around the y-axis
by the method of cylindrical shells.

\(\displaystyle \L V\;=\;2\pi\int^{\;\;\;b}_a x\cdot f(x)\,dx\)

. . . \(\displaystyle \L= \;2\pi\int x(2\cdot\cos x)\,dx\)

. . . \(\displaystyle \L= \;4\pi\int \underbrace{x\cdot\cos x}\,dx\)
. . . . . . . . . . . "by parts"

. . \(\displaystyle \begin{array}{ccc}u\,=\,x & \;\;\; & dv\,=\,\cos x\,dx \\
du\,=\,dx & \;\;\; &v\,=\,\sin x \end{array}\)

Then: \(\displaystyle \L\:V \;=\;4\pi\left[x\cdot\sin x\:-\:\int\sin x\,dx\right]^{\frac{\pi}{2}}_0\)

Got it?

 

[galactus]