Solids of Revolution

The Shell Method

The graph below shows the region R bounded between the graph y = sin(x) (with 0 ≤ x ≤ π) and the x-axis (red). Select different rectangular slices of region R by adjusting the k-value. Rotate this slice around the y-axis by increasing the θ-value. [br] Setting θ = 3.14 gives you the option to view the solid of revolution that results when region R is rotated about the y-axis (blue).
This sketch is designed to help you visualize an increment of volume ΔV represented by a single cylindrical shell. ΔV can be approximated by 2πrh·Δr, an approximation that gets more accurate as Δr gets closer to zero. [br] In this example, r = x, Δr = Δx, and h = sin(x), so ΔV ≈ 2πx·sin(x)·Δx. To obtain the actual volume, we integrate the differential dV = 2πx·sin(x)·dx from x=0 to x=π.