Glossary

Axis of Rotation (y=b or x=a)

Criticality: 3

The specific horizontal line (y=b) or vertical line (x=a) around which a 2D region is revolved to generate a 3D solid.

Example:

If you rotate the region under $y=\sqrt{x}$ around the line x=4, then x=4 is your axis of rotation.

Circular Cross-sections

Criticality: 2

The two-dimensional shapes formed when a solid of revolution is sliced perpendicular to its axis of rotation, always resulting in circles for the disc method.

Example:

When you slice a cone horizontally, each slice reveals a circular cross-section.

Disc Method

Criticality: 3

A calculus technique used to find the volume of a solid of revolution by summing the volumes of infinitesimally thin circular discs perpendicular to the axis of rotation.

Example:

To find the volume of a sphere, you can rotate a semicircle around the x-axis and apply the Disc Method.

Integral (for Volume)

Criticality: 3

A mathematical tool used to sum an infinite number of infinitesimally small quantities, representing the total volume by accumulating the areas of cross-sections along an axis.

Example:

The integral $\int_{a}^{b} A(x) dx$ calculates the total volume by summing the area A(x) of each slice from x=a to x=b.

Intersections (for Limits)

Criticality: 2

The points where functions or a function and an axis meet, which often serve as the natural boundaries or limits of integration for volume calculations.

Example:

To find the volume of the region bounded by $y=x^2$ and $y=4$ revolved around an axis, you'd first find their intersections at $x=\pm 2$ .

Radius (r) for Disc Method

Criticality: 3

The distance from the axis of rotation to the outer edge of the function being revolved, which determines the size of each circular disc.

Example:

If revolving $y=x^2$ around $y=-1$ , the radius of a disc at a given x-value would be $x^2 - (-1) = x^2+1$ .

Revolving Around Other Axes

Criticality: 3

The process of rotating a 2D region around a horizontal line (y=b) or a vertical line (x=a) that is not the standard x or y-axis to form a 3D solid.

Example:

Calculating the volume of a donut shape by revolving a circle around a vertical line like x=5 is an example of revolving around another axis.

Upper and Lower Limits of Integration

Criticality: 3

The boundary values (c and d) that define the specific interval over which the volume integral is calculated, representing the extent of the solid along the axis of integration.

Example:

For a solid formed by revolving a region from x=0 to x=2, the upper and lower limits of integration would be 0 and 2.

Volume Formula (Disc Method)

Criticality: 3

The integral expression used to calculate the volume of a solid of revolution using the disc method, typically $\int_{c}^{d}\pi (f(x)-b)^2 dx$ for horizontal axes or $\int_{c}^{d}\pi (f(y)-a)^2 dy$ for vertical axes.

Example:

The volume formula for revolving $y=x^2$ around $y=-1$ from $x=0$ to $x=2$ is $\int_{0}^{2}\pi (x^2 - (-1))^2 dx$ .