Final answer:
To find the volume of the solid generated by revolving the region bounded by y=2sinx and y=0 about the x-axis, we can use the method of cylindrical shells can be calculated as 8π cubic units.
Step-by-step explanation:
To find the volume of the solid generated by revolving the region bounded by y=2sinx and y=0 about the x-axis, we can use the method of cylindrical shells.
- First, we will calculate the limits of integration. Equate y=2sinx to zero to find the points of intersection: 2sinx=0. This occurs at x=0 and x=π.
- The radius of each cylindrical shell is y=2sinx, and the height is dx. So, the circumference of each shell is 2πy=4πsinx.
- The volume of each shell is given by dV=2πydx=4πsinxdx.
- To find the total volume, integrate the volume of each shell over the interval [0,π]:
Volume = ∫(4πsinx)dx, limits of integration: 0 to π
Integrating this, we get:
Volume = [-4πcosx] from 0 to π
Volume = 8π