washermethode

Washer method, or the washer method, is a technique in integral calculus used to compute the volume of solids of revolution. It generalizes the disk method by allowing a hole in the solid, which occurs when the region being revolved is bounded by two curves.

The basic setup for revolution about the x-axis: Suppose a region R lies between y = f(x) and y = g(x) on the interval a ≤ x ≤ b, with f(x) ≥ g(x) ≥ 0. When R is revolved about the x-axis, a vertical slice at position x forms a washer with outer radius R(x) = f(x) and inner radius r(x) = g(x). The volume is V = π ∫_a^b [f(x)^2 − g(x)^2] dx. If g(x) ≡ 0, this reduces to the disk method.

For revolution about the y-axis, the corresponding setup uses horizontal slices. If the region is described by x = F(y) and x = G(y) for y in [c, d], the washers have outer radius F(y) and inner radius G(y), giving V = π ∫_c^d [F(y)^2 − G(y)^2] dy.

Key points: the radii are measured from the axis of revolution to the outer and inner boundaries of the region. The washer method is particularly useful when the solid has a hole or a cavity. It is related to the disk method, which is the special case with no hole (inner radius zero). Splitting the region or using the shell method can help when a single expression in x or y is not available.