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

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

$\int_{c}^{d}\pi (R(x)^2 - r(x)^2) dx$, 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.

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

Disc: Solid has no hole, single radius. Washer: Solid has a hole, inner and outer radii.

x-axis: Integrate with respect to x, radii are vertical. y-axis: Integrate with respect to y, radii are horizontal.

y=b: Horizontal axis, radii are vertical, functions in terms of x. x=a: Vertical axis, radii are horizontal, functions in terms of y.

Washer: Integrate perpendicular to the axis, uses radii. Shell: Integrate parallel to the axis, uses height and radius.

Outer: Distance from axis to the farthest curve. Inner: Distance from axis to the closest curve.

x: Used for horizontal axes, functions in terms of x. y: Used for vertical axes, functions in terms of y.

Horizontal: $V = \int \pi [(f(x)-b)^2 - (g(x)-b)^2] dx$. Vertical: $V = \int \pi [(f(y)-a)^2 - (g(y)-a)^2] dy$.

Disc/Washer: Easier when axis is perpendicular to the slicing. Shell: Easier when axis is parallel to the slicing.

Simple: Easier to find intersection points and integrate. Complex: May require calculator for intersection and more advanced integration techniques.

Constant: Volume = Area * Height. Variable: Volume = Integral of Area function over the interval.

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).