The
**washer method**refers 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:

$$\mathrm{V}={\int}_{\mathrm{a}}^{\mathrm{b}}\Pi \left\{{\left[f\left(x\right)\right]}^{2}-{\left[g\left(x\right)\right]}^{2}\right\}dx$$

If the functions revolve around the 'y' axis, the radii are the 'y' values of the function, or:

$$\mathrm{V}={\int}_{\mathrm{a}}^{\mathrm{b}}\Pi \left\{{\left[f\left(y\right)\right]}^{2}-{\left[g\left(y\right)\right]}^{2}\right\}dy$$

In this example, we look for the volume of a solid whose shape is bound by a functions y = x2 + 2, and the function y = x + 4.

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

So the volume V of a solid generated by revolving a region bounded by

$$\mathrm{V}={\int}_{\mathrm{a}}^{\mathrm{b}}\Pi \left\{{\left[f\left(x\right)\right]}^{2}-{\left[g\left(x\right)\right]}^{2}\right\}dx$$

If the functions revolve around the 'y' axis, the radii are the 'y' values of the function, or:

$$\mathrm{V}={\int}_{\mathrm{a}}^{\mathrm{b}}\Pi \left\{{\left[f\left(y\right)\right]}^{2}-{\left[g\left(y\right)\right]}^{2}\right\}dy$$

In this example, we look for the volume of a solid whose shape is bound by a functions y = x

Revolving solid planes - washer method

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