How to find volume with the washer method

If the region stays away from its axis of rotation, a cross section can have a hole. We can find its area by subtracting the smaller disk from the larger one. Let’s find those two radii and use them to set up the volume.

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Measure both radii from the axis

$A=\pi (R^2-r^2)$

The area of the washer is the outer disk area minus the hole.

Find where the bounding curves meet

Rotate the region between $y=x^2+2$ and $y=x+4$ about the $x$-axis. The curves meet at $x=-1$ and $x=2$ because $x^2-x-2=(x+1)(x-2)=0$.

Between those points, the line lies above the parabola. Both are above the axis, so the outer radius is $x+4$ and the inner radius is $x^2+2$.

$\begin{array}{l}V=\pi {\int }_{-1}^2\left(\right(x+4{)}^2-(x^2+2{)}^2)dx\\ =\pi {\int }_{-1}^2(-x^4-3x^2+8x+12)dx\\ =\frac{162\pi }{5}\end{array}$

Use horizontal slices around a vertical axis

A vertical axis calls for horizontal washers, so write both radii in terms of $y$ and integrate with respect to $y$. Measure each distance from the same axis, even when the axis is shifted.

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