Editing Disc integration (section)

===Washer method===
To obtain a hollow solid of revolution (the “washer method”), the procedure would be to take the volume of the inner solid of revolution and subtract it from the volume of the outer solid of revolution. This can be calculated in a single integral similar to the following:

:<math>\pi\int_a^b\left(R_\mathrm{O}(x)^2 - R_\mathrm{I}(x)^2\right)\,dx</math>

where {{math|''R''<sub>O</sub>(''x'')}} is the function that is furthest from the axis of rotation and {{math|''R''<sub>I</sub>(''x'')}} is the function that is closest to the axis of rotation. For example, the next figure shows the rotation along the {{mvar|x}}-axis of the red "leaf" enclosed between the square-root and quadratic curves: 
[[File:Solid of revolution.gif|thumb|Rotation about x-axis]]
The volume of this solid is:
:<math>\pi\int_0^1\left(\left(\sqrt{x}\right)^2 - \left(x^2\right)^2 \right)\,dx\,.</math>

One should take caution not to evaluate the square of the difference of the two functions, but to evaluate the difference of the squares of the two functions.

:<math>R_\mathrm{O}(x)^2 - R_\mathrm{I}(x)^2 \neq \left(R_\mathrm{O}(x) - R_\mathrm{I}(x)\right)^2</math>

(This formula only works for revolutions about the {{mvar|x}}-axis.)

To rotate about any horizontal axis, simply subtract from that axis from each formula. If {{mvar|h}} is the value of a horizontal axis, then the volume equals

:<math>\pi\int_a^b\left(\left(h-R_\mathrm{O}(x)\right)^2 - \left(h-R_\mathrm{I}(x)\right)^2\right)\,dx\,.</math>

For example, to rotate the region between {{math|''y'' {{=}} −2''x'' + ''x''<sup>2</sup>}} and {{math|''y'' {{=}} ''x''}} along the axis {{math|''y'' {{=}} 4}}, one would integrate as follows:

:<math>\pi\int_0^3\left(\left(4-\left(-2x+x^2\right)\right)^2 - (4-x)^2\right)\,dx\,.</math>

The bounds of integration are the zeros of the first equation minus the second. Note that when integrating along an axis other than the {{mvar|x}}, the graph of the function which is furthest from the axis of rotation may not be obvious. In the previous example, even though the graph of {{math|''y'' {{=}} ''x''}} is, with respect to the x-axis, further up than the graph of {{math|''y'' {{=}} −2''x'' + ''x''<sup>2</sup>}}, with respect to the axis of rotation the function {{math|''y'' {{=}} ''x''}} is the inner function: its graph is closer to {{math|''y'' {{=}} 4}} or the equation of the axis of rotation in the example.

The same idea can be applied to both the {{mvar|y}}-axis and any other vertical axis. One simply must solve each equation for {{mvar|x}} before one inserts them into the integration formula.