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.

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

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

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

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

The distance represents the difference between the outer and inner radii, which determines the area of the washer.

It's a horizontal or vertical line that the region is rotated around to form the solid.

They are the points where the curves intersect, defining the limits of integration.

A larger volume because the washer has a greater area.

It's the function that is farther away from the axis of revolution.

The shape dictates the overall form of the solid and the variation in the radii of the washers.

It means the region is unbounded, or there might be a different region specified for the problem.

Visually approximate the area of the washers and integrate mentally over the interval.

If the region is symmetric, you can integrate over half the interval and multiply by two.

It changes the orientation of the solid and the way the radii are measured from the axis.