carlosgrahm said:
The washer method and cylindrical shells method are two different approaches used in Calculus 2 to find the volume of solids of revolution. The main difference between them is the shape of the cross-sections used to create the solid. The washer method uses circular cross-sections, while the cylindrical shells method uses rectangular cross-sections.
The choice between the washer method and cylindrical shells method depends on the shape of the solid of revolution. If the solid has a hole or empty space in the middle, the washer method should be used. If the solid does not have a hole, the cylindrical shells method should be used.
For the washer method, the integral is set up as follows: ∫a^b π(R^2 - r^2)dx, where R is the outer radius, r is the inner radius, and dx represents the width of the solid. For the cylindrical shells method, the integral is set up as follows: ∫a^b 2πrhdx, where r is the distance from the axis of rotation to the edge of the solid, h is the height of the shell, and dx represents the width of the solid.
Both the washer method and cylindrical shells method are equally accurate in finding the volume of solids of revolution. The choice between the two methods depends on the shape of the solid and the ease of setting up the integral.
Yes, there are other methods such as the disk method and the method of cross-sections. These methods involve using different shapes for the cross-sections and may be more suitable for certain types of solids. It is important to understand the different methods and choose the most appropriate one for each specific problem.