Glossary

Axis of Rotation

Criticality: 3

The line around which a two-dimensional region is revolved to create a three-dimensional solid.

Example:

If you're rotating a region around the line $y=1$ , then $y=1$ is your axis of rotation.

Disc Method

Criticality: 2

A calculus technique used to find the volume of a solid of revolution when the region being rotated extends directly to the axis of rotation, forming solid circular cross-sections.

Example:

To find the volume of a solid sphere, you could use the Disc Method by rotating a semicircle around the x-axis.

Inner Radius ($g(x)$)

Criticality: 3

In the Washer Method, this is the function that is nearer to the axis of rotation within the specified interval, determining the smaller radius of the washer.

Example:

When revolving the region between $y=x^2$ and $y=x$ around the x-axis for $x \in [0,1]$ , $y=x^2$ would be the inner radius function as it is closer to the x-axis.

Intersection Points

Criticality: 2

The coordinates where two or more functions meet, often used to determine the upper and lower bounds for integration in volume problems.

Example:

To set up the integral for the volume between $y=x^2$ and $y=x$ , you first need to find their intersection points at $x=0$ and $x=1$ .

Lower Bound (c)

Criticality: 2

The starting x- or y-value that defines the beginning of the region being rotated, serving as the lower limit of integration.

Example:

If a region is bounded by $x=0$ and $x=2$ , then $x=0$ is the lower bound for your integral.

Outer Radius ($f(x)$)

Criticality: 3

In the Washer Method, this is the function that is farther from the axis of rotation within the specified interval, determining the larger radius of the washer.

Example:

When revolving the region between $y=x^2$ and $y=x$ around the x-axis for $x \in [0,1]$ , $y=x$ would be the outer radius function because it is farther from the x-axis.

Upper Bound (d)

Criticality: 2

The ending x- or y-value that defines the end of the region being rotated, serving as the upper limit of integration.

Example:

If a region is bounded by $x=0$ and $x=2$ , then $x=2$ is the upper bound for your integral.

Volume of a Solid of Revolution

Criticality: 3

The three-dimensional space occupied by a solid formed by rotating a two-dimensional region around an axis.

Example:

Calculating the volume of a solid of revolution can help engineers determine the capacity of a cylindrical tank or the material needed for a specific part.

Washer (geometric shape)

Criticality: 1

A two-dimensional shape resembling a flat ring, formed by a larger circle with a smaller concentric circle removed from its center.

Example:

When you slice through a solid generated by the Washer Method, each cross-section will look like a washer.

Washer Method

Criticality: 3

A calculus technique used to find the volume of a solid of revolution when the region being rotated has a hole in the middle, formed by revolving the area between two functions around an axis.

Example:

To calculate the volume of a solid shaped like a ring or a thick pipe, you would apply the Washer Method by rotating the area between two curves.

Glossary

Axis of Rotation

Criticality: 3

The line around which a two-dimensional region is revolved to create a three-dimensional solid.

Example:

If you're rotating a region around the line $y=1$ , then $y=1$ is your axis of rotation.

Disc Method

Criticality: 2

A calculus technique used to find the volume of a solid of revolution when the region being rotated extends directly to the axis of rotation, forming solid circular cross-sections.

Example:

To find the volume of a solid sphere, you could use the Disc Method by rotating a semicircle around the x-axis.

Inner Radius ($g(x)$)

Criticality: 3

In the Washer Method, this is the function that is nearer to the axis of rotation within the specified interval, determining the smaller radius of the washer.

Example:

When revolving the region between $y=x^2$ and $y=x$ around the x-axis for $x \in [0,1]$ , $y=x^2$ would be the inner radius function as it is closer to the x-axis.

Intersection Points

Criticality: 2

The coordinates where two or more functions meet, often used to determine the upper and lower bounds for integration in volume problems.

Example:

To set up the integral for the volume between $y=x^2$ and $y=x$ , you first need to find their intersection points at $x=0$ and $x=1$ .

Lower Bound (c)

Criticality: 2

The starting x- or y-value that defines the beginning of the region being rotated, serving as the lower limit of integration.

Example:

If a region is bounded by $x=0$ and $x=2$ , then $x=0$ is the lower bound for your integral.

Outer Radius ($f(x)$)

Criticality: 3

In the Washer Method, this is the function that is farther from the axis of rotation within the specified interval, determining the larger radius of the washer.

Example:

When revolving the region between $y=x^2$ and $y=x$ around the x-axis for $x \in [0,1]$ , $y=x$ would be the outer radius function because it is farther from the x-axis.

Upper Bound (d)

Criticality: 2

The ending x- or y-value that defines the end of the region being rotated, serving as the upper limit of integration.

Example:

If a region is bounded by $x=0$ and $x=2$ , then $x=2$ is the upper bound for your integral.

Volume of a Solid of Revolution

Criticality: 3

The three-dimensional space occupied by a solid formed by rotating a two-dimensional region around an axis.

Example:

Calculating the volume of a solid of revolution can help engineers determine the capacity of a cylindrical tank or the material needed for a specific part.

Washer (geometric shape)

Criticality: 1

A two-dimensional shape resembling a flat ring, formed by a larger circle with a smaller concentric circle removed from its center.

Example:

When you slice through a solid generated by the Washer Method, each cross-section will look like a washer.

Washer Method

Criticality: 3

A calculus technique used to find the volume of a solid of revolution when the region being rotated has a hole in the middle, formed by revolving the area between two functions around an axis.

Example:

To calculate the volume of a solid shaped like a ring or a thick pipe, you would apply the Washer Method by rotating the area between two curves.