This single equation does not define a bounded region. Was there some other line or curve given, such as the y-axis (x= 0)?whatlifeforme said:Homework Statement
use the shell method to find the volumes of the solids generated by revolving the region about the indicated axis.
Homework Equations
x=18(y^2 - y^3) about line y=8/5.i
The Attempt at a Solution
2∏∫ (0 to 1) (8/5 - y) (18y^2 - 18y^3)
whatlifeforme said:2∏∫ (0 to 1) (8/5 - y) (18y^2 - 18y^3) dy
The volume shell method is a mathematical technique used to calculate the volume of a solid of revolution. It involves slicing the solid into thin cylindrical shells and using the formula V = 2πrh to find the volume of each shell, then adding up the volumes of all the shells to get the total volume.
The volume shell method is typically used when the solid being rotated around an axis is not a simple geometric shape, such as a cone or cylinder. It is also useful when the axis of rotation is not along the x- or y-axis.
The volume disk method involves slicing a solid of revolution into thin disks and using the formula V = πr^2h to calculate the volume of each disk. The volume shell method, on the other hand, involves slicing the solid into thin cylindrical shells and using the formula V = 2πrh to calculate the volume of each shell.
1. Identify the axis of rotation.2. Determine the limits of integration.3. Express the function in terms of the variable of integration.4. Set up the integral using the formula V = 2πrh.5. Integrate to find the total volume.
Yes, the volume shell method can be used for solids with holes as long as the holes do not intersect with the axis of rotation. In this case, the volume of the holes would need to be subtracted from the total volume calculated using the volume shell method.