Glossary

Area of a Cross-Section (A(x))

Criticality: 3

A function that describes the area of a two-dimensional slice of a solid at a specific point along an axis.

Example:

For a solid with circular cross-sections, the Area of a Cross-Section (A(x)) might be $\pi r(x)^2$ , where $r(x)$ is the radius at a given $x$ .

Bounds of Integration

Criticality: 3

The upper and lower limits of the definite integral, representing the interval over which the cross-sections are summed to find the total volume.

Example:

For a solid extending from $x=0$ to $x=5$ , the bounds of integration would be 0 and 5.

Definite Integral

Criticality: 3

An integral with upper and lower limits, representing the net accumulation of a quantity over a specific interval.

Example:

To find the total volume of a solid, you evaluate a definite integral of its cross-sectional area function.

Height (h)

Criticality: 2

The other dimension of a rectangular cross-section, which may be a constant value or a function of x or y.

Example:

For a solid with rectangular cross-sections where the height is always twice the width, the height (h) would be $2w$ .

Lower Curve

Criticality: 2

In the context of finding the side length or width of a cross-section, it refers to the function that defines the bottom boundary of the base region.

Example:

If a region is bounded by $y = \sin(x)$ and $y = x^2$ , for $x \in (0,1)$ , $y = x^2$ would be the lower curve.

Perpendicular to the x-axis

Criticality: 2

Describes cross-sections that are oriented such that their plane is at a right angle to the x-axis, meaning the integral will be with respect to x (dx).

Example:

If you slice a solid parallel to the yz-plane, the cross-sections are perpendicular to the x-axis.

Perpendicular to the y-axis

Criticality: 2

Describes cross-sections that are oriented such that their plane is at a right angle to the y-axis, meaning the integral will be with respect to y (dy).

Example:

When calculating the volume of a solid by slicing it horizontally, the cross-sections are often perpendicular to the y-axis.

Rectangular Cross Sections

Criticality: 3

A type of cross-section where each slice of the solid forms a rectangle, with its area given by $w \cdot h$.

Example:

A solid whose base is an ellipse and whose slices are rectangles of constant height would involve rectangular cross sections.

Side (s)

Criticality: 2

The length of one side of a square cross-section, typically found by calculating the distance between two bounding curves.

Example:

If a square cross-section's base extends from $y=x^2$ to $y=\sqrt{x}$ , then its side (s) length is $\sqrt{x} - x^2$ .

Solid with Known Cross-Sections

Criticality: 3

A three-dimensional object whose volume can be determined by knowing the shape and area function of its cross-sections.

Example:

A pyramid can be viewed as a solid with known cross-sections where each horizontal slice is a square.

Square Cross Sections

Criticality: 3

A type of cross-section where each slice of the solid forms a square, with its area given by $s^2$.

Example:

If the base of a solid is a circle and its vertical slices are squares, you're dealing with square cross sections.

Thickness (dx/dy)

Criticality: 2

An infinitesimally small dimension representing the depth of each cross-sectional slice, used in the integral to accumulate volume.

Example:

In the formula $V = \int A(x) \textbf{dx}$ , the thickness (dx) indicates that the slices are perpendicular to the x-axis.

Upper Curve

Criticality: 2

In the context of finding the side length or width of a cross-section, it refers to the function that defines the top boundary of the base region.

Example:

If a region is bounded by $y = \sin(x)$ and $y = x^2$ , for $x \in (0,1)$ , $y = \sin(x)$ would be the upper curve.

Volumes with Cross Sections

Criticality: 3

A method to calculate the volume of a three-dimensional object by integrating the area of its two-dimensional cross-sections over a given interval.

Example:

Imagine slicing a loaf of bread; each slice is a cross section, and summing their areas with their thickness gives the total volume.

Width (w)

Criticality: 2

One dimension of a rectangular cross-section, often determined by the distance between two curves or a curve and an axis.

Example:

If a rectangular cross-section spans from $x=g(y)$ to $x=h(y)$ , its width (w) would be $h(y) - g(y)$ .