The washer methodrefers to the technique used to find the volume formed by a shape that revolves around an axis, BUT the axis itself is NOT one of the boundaries of the plane that is revolving.
In these cases, the problem is initially approached the same way as the 'disk' problem, but we have to take into account the hole that is cut out the disk; the hole is formed by the smaller circle that is formed by the offset of the function from the 'x' or 'y' axis.
As before, the volumes of these disks are found by multiplying the area of each circle (Πr 2). The value 'r' of the radius of the circles is the value of 'y' for each function ( f(x)and g(x)). The total volume is found by subtracting the volume formed by the 'inner' washer from the volume formed by the 'outer' washer.
So the volume V of a solid generated by revolving a region bounded by y = f(x)and the function y = g(x)over the interval [a,b], assuming f(x) ≥ g(x)and that the function revolves around the 'x' axis, is found by:
$V={\int }_{a}^{b}\Pi \left\{{\left[f\left(x\right)\right]}^{2}-{\left[g\left(x\right)\right]}^{2}\right\}dx$
If the functions revolve around the 'y' axis, the radii are the 'y' values of the function, or:
$V={\int }_{a}^{b}\Pi \left\{{\left[f\left(y\right)\right]}^{2}-{\left[g\left(y\right)\right]}^{2}\right\}dy$
In this example, we look for the volume of a solid whose shape is bound by a functions y = x 2+ 2, and the function y = x + 4.
Revolving solid planes - washer method

The URL listed above can be inserted into your postings or e-mails; it will point to the text or area of the graph that you have selected.

• Drag your mouse across any text in the green problem area, or
• Drag your mouse to form a selection box on the green graph area
• Open the e-Blink tab to display this tab.
• Copy the text above to your clipboard
• When editing a note to your community or editing your e-mail, select some text and press the link button.