The disc method and shell method are two different techniques used to find the volume of a solid of revolution. The disc method involves slicing the solid into discs perpendicular to the axis of revolution, while the shell method involves slicing the solid into cylindrical shells parallel to the axis of revolution.
The choice between disc and shell method depends on the shape of the solid of revolution. The disc method is typically used when the cross-sections of the solid are circles, while the shell method is used when the cross-sections are rectangles or squares.
The setup for both disc and shell integrals involves finding the radius, height, and thickness of each disc or shell. For the disc method, the radius is the distance from the axis of revolution to the disc, and the height is the thickness of the disc. For the shell method, the radius is the distance from the axis of revolution to the center of the shell, and the height is the height of the shell. The thickness is the same for both methods and represents the change in the variable of integration.
Yes, it is possible to use both disc and shell methods to find the volume of the same solid. However, the choice of which method to use may affect the complexity of the integral and the amount of work required to solve it.
Both disc and shell methods have limitations, especially when the cross-sections of the solid are not simple shapes. In those cases, other integration methods such as the washer method or the method of cylindrical shells may be more appropriate.