$\int_{c}^{d}\pi [(f(x)-b)^2 - (g(x)-b)^2] dx$

$\int_{c}^{d}\pi [(f(y)-a)^2 - (g(y)-a)^2] dy$

$\int_{c}^{d}\pi (R(x)^2 - r(x)^2) dx$, where R(x) is the outer radius and r(x) is the inner radius.

$A = \pi r^2$

$V = \int_{a}^{b} A(x) dx$, where A(x) is the area of the cross-section at x.

$R(x) = |f(x) - b|$, where f(x) is the function farther from the axis of revolution.

$r(x) = |g(x) - b|$, where g(x) is the function closer to the axis of revolution.

$\int_{a}^{b} \pi (radius)^2 dx$

Find the points of intersection between the curves, these x or y values will be your limits.

$V = \int_{c}^{d} \pi [(R(y))^2 - (r(y))^2] dy$

A method to find the volume of a solid of revolution by subtracting the volume of a smaller inner solid from a larger outer solid.

A 3D shape formed by rotating a 2D curve around an axis.

The distance from the axis of revolution to the outer boundary of the region being rotated.

The distance from the axis of revolution to the inner boundary (hole) of the region being rotated.

The x or y values that define the interval over which the solid is formed.

The line around which a 2D region is rotated to create a 3D solid.

To find the volume of solids of revolution that have a hole in the middle.

A slice of the 3D solid perpendicular to the axis of revolution.

The disc method is a special case of the washer method where the inner radius is zero.

$\pi$ is used to calculate the area of the circular cross-sections (washers).

Divide the solid into thin slices, approximate the volume of each slice, and sum the volumes using integration.

The Disc Method is used when the solid has no hole, while the Washer Method is used when there is a hole.

The axis of revolution determines the shape of the cross-sections and how the radius is calculated.

The outer radius function must be subtracted by the inner radius function to get a positive volume.

Use absolute values to ensure the radius is always positive, especially when revolving around axes other than x or y.

Graphing helps visualize the solid, identify the outer and inner radii, and determine the correct limits of integration.

The Washer Method extends the concept of area between curves to three dimensions by revolving the area around an axis.

Integration sums up the infinitesimal volumes of the washers to find the total volume of the solid.

Sketch the region, identify the axis of revolution, determine the inner and outer radii, find the limits of integration, and set up the integral.

It changes the expressions for the inner and outer radii and may require integrating with respect to y instead of x.