Which method is typically used to find the volume of a solid of revolution around the y-axis using vertical slices?

Which method is typically used to find the volume of a solid of revolution around the y-axis using vertical slices?

• A. Washer method
• B. Shells method
• C. Disk method
• D. Pappus' theorem

Answer: (B)

Think about rotating a vertical slice around a vertical axis. That slice, when revolved about the y-axis, makes a thin cylindrical shell with radius equal to its x-distance from the axis and height equal to the vertical extent of the region (top minus bottom, or f(x) − g(x) if two curves bound it). The volume of that shell is 2π times radius times height times thickness, so dV = 2π x [f(x) − g(x)] dx. Integrating from the left to right x-limits gives V = ∫ from a to b 2π x [f(x) − g(x)] dx. This shell method is the natural choice when the axis of rotation is vertical and you’re using vertical slices. Disk or washer methods would require horizontal slices for rotation around a vertical axis, and Pappus’ theorem is a different approach that’s not the standard slice-by-slice method here.

Think about rotating a vertical slice around a vertical axis. That slice, when revolved about the y-axis, makes a thin cylindrical shell with radius equal to its x-distance from the axis and height equal to the vertical extent of the region (top minus bottom, or f(x) − g(x) if two curves bound it). The volume of that shell is 2π times radius times height times thickness, so dV = 2π x [f(x) − g(x)] dx. Integrating from the left to right x-limits gives V = ∫ from a to b 2π x [f(x) − g(x)] dx. This shell method is the natural choice when the axis of rotation is vertical and you’re using vertical slices. Disk or washer methods would require horizontal slices for rotation around a vertical axis, and Pappus’ theorem is a different approach that’s not the standard slice-by-slice method here.