What is the volume of the solid formed by the line?

The graph of #y=2sin((pix)/2)# from x = 0 to x = 2 is rotated around the y-axis to form a solid figure. Find the volume of the solid.enter image source here

1 Answer
May 24, 2018

#color(blue)[V=piint_0^2(2sin((pix)/2))^2*dx=4pi]#

Explanation:

we will use Disk method to calculate the volume of the solid generated from rotating the curve around #"x-axis"#

the interval of the integral of volume #x in [0,2]#

The volume of the solid between the curve and the axis is given by:

#V=piint_a^by^2*dx#

#V=piint_0^2(2sin((pix)/2))^2*dx#

#V=piint_0^2(4sin^2((pix)/2))*dx#

#V=4piint_0^2(1/2-1/2*cos(pix))*dx#

#V=2piint_0^2(1-cos(pix))*dx#

#=2pi[x-1/pisin(pix)]_0^2=[(2-1/pi*sin2pi)-(-1/pisin0)]#

#2-0+0=4pi#