Question

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.

Answer 1

`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`