The washer method is a technique used to find the volume of a solid of revolution when the region being revolved around an axis creates a hollow shape. It involves integrating the area of washers, which are disks with holes in the center.
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The washer method is used when the solid has a gap or hole along the axis of rotation.
The formula for the volume using the washer method is $$V = \pi \int_{a}^{b} [R(x)^2 - r(x)^2] \, dx$$ where $R(x)$ and $r(x)$ are the outer and inner radii, respectively.
Ensure that $R(x) \geq r(x)$ over the interval [a, b] to avoid negative volumes.
If rotating around the y-axis, express $x$ as a function of $y$ and integrate with respect to $y$.
The washer method can be applied to both horizontal and vertical axes of rotation.
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Related terms
Disk Method: A technique for finding volumes by slicing perpendicular to an axis of rotation, creating solid disks without holes.
Solid of Revolution: A three-dimensional object obtained by rotating a two-dimensional region around an axis.
Cylindrical Shells Method: A technique for finding volumes by slicing parallel to an axis of rotation, forming cylindrical shells.