Glossary

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

Criticality: 3

A function that represents the area of a two-dimensional shape (the cross-section) at a specific point along the axis perpendicular to which the cross-sections are taken.

Example:

If a solid has square cross-sections, its Area of a Cross-Section (A(x)) might be $(y_{upper} - y_{lower})^2$ , where $y_{upper} - y_{lower}$ is the side length of the square.

Bounds of Integration (a, b)

Criticality: 3

The lower and upper limits of the definite integral, representing the interval along the axis perpendicular to the cross-sections over which the volume is calculated.

Example:

When finding the volume of a solid whose base is bounded by $y=x^2$ and $y=x$ , the bounds of integration (a, b) would be from $x=0$ to $x=1$ , where the curves intersect.

Equilateral Triangle Cross Sections

Criticality: 2

Cross-sections of a solid that are equilateral triangles, meaning all three sides are equal in length. The area formula is $\frac{\sqrt{3}}{4}s^2$, where $s$ is the side length.

Example:

Imagine a solid whose base is a region bounded by curves, and every slice perpendicular to the x-axis forms an Equilateral Triangle Cross Section; you'd use $\frac{\sqrt{3}}{4}s^2$ for $A(x)$ .

Radius (r)

Criticality: 3

In the context of semicircular cross-sections, 'r' represents the radius of the semicircle, which is half of its diameter. The diameter is typically determined by the distance between two bounding curves of the solid's base.

Example:

If a solid has semicircular cross-sections, the radius (r) of each semicircle is half the distance between the top and bottom curves of the base region.

Right Isosceles Triangle Cross Sections

Criticality: 2

Cross-sections of a solid that are right-angled isosceles triangles, meaning they have a right angle and two equal sides. The area formula is $\frac{1}{2}s^2$, where $s$ is the length of the two matching sides.

Example:

If a solid has a circular base and its cross-sections perpendicular to the diameter are Right Isosceles Triangle Cross Sections with the hypotenuse on the base, you'd need to relate the leg length 's' to the circle's equation.

Semicircular Cross Sections

Criticality: 2

Cross-sections of a solid that are semicircles, meaning half of a circle. The area formula is $\frac{1}{2}\pi r^2$, where $r$ is the radius of the semicircle.

Example:

A solid with a parabolic base might have Semicircular Cross Sections perpendicular to the x-axis, where the diameter of each semicircle is the distance between the two branches of the parabola.

Side Length (s)

Criticality: 3

In the context of triangular cross-sections, 's' represents the length of a side of the triangle, often determined by the distance between two bounding curves of the solid's base.

Example:

For a solid with square cross-sections, the side length (s) of each square might be given by the difference between the upper and lower functions, like $y_{top} - y_{bottom}$ .

Volume Integral

Criticality: 3

The definite integral $V = \int_a^b A(x) dx$ (or $V = \int_c^d A(y) dy$), used to calculate the total volume of a solid by summing the infinitesimal volumes of its cross-sections.

Example:

To find the volume of a solid with known cross-sections, you would set up and evaluate a Volume Integral like $\int_0^2 (x^2) dx$ if the cross-sections were squares with side length $x$ .

Volumes with Cross Sections

Criticality: 3

A method to calculate the volume of a 3D solid by integrating the area of its 2D cross-sections taken perpendicular to an axis over a given interval.

Example:

To find the volume of a solid whose base is a circle and whose cross-sections perpendicular to the x-axis are squares, you would use the Volumes with Cross Sections method.

Glossary

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

Criticality: 3

A function that represents the area of a two-dimensional shape (the cross-section) at a specific point along the axis perpendicular to which the cross-sections are taken.

Example:

If a solid has square cross-sections, its Area of a Cross-Section (A(x)) might be $(y_{upper} - y_{lower})^2$ , where $y_{upper} - y_{lower}$ is the side length of the square.

Bounds of Integration (a, b)

Criticality: 3

The lower and upper limits of the definite integral, representing the interval along the axis perpendicular to the cross-sections over which the volume is calculated.

Example:

When finding the volume of a solid whose base is bounded by $y=x^2$ and $y=x$ , the bounds of integration (a, b) would be from $x=0$ to $x=1$ , where the curves intersect.

Equilateral Triangle Cross Sections

Criticality: 2

Cross-sections of a solid that are equilateral triangles, meaning all three sides are equal in length. The area formula is $\frac{\sqrt{3}}{4}s^2$, where $s$ is the side length.

Example:

Imagine a solid whose base is a region bounded by curves, and every slice perpendicular to the x-axis forms an Equilateral Triangle Cross Section; you'd use $\frac{\sqrt{3}}{4}s^2$ for $A(x)$ .

Radius (r)

Criticality: 3

Example:

If a solid has semicircular cross-sections, the radius (r) of each semicircle is half the distance between the top and bottom curves of the base region.

Right Isosceles Triangle Cross Sections

Criticality: 2

Example:

Semicircular Cross Sections

Criticality: 2

Cross-sections of a solid that are semicircles, meaning half of a circle. The area formula is $\frac{1}{2}\pi r^2$, where $r$ is the radius of the semicircle.

Example:

A solid with a parabolic base might have Semicircular Cross Sections perpendicular to the x-axis, where the diameter of each semicircle is the distance between the two branches of the parabola.

Side Length (s)

Criticality: 3

In the context of triangular cross-sections, 's' represents the length of a side of the triangle, often determined by the distance between two bounding curves of the solid's base.

Example:

For a solid with square cross-sections, the side length (s) of each square might be given by the difference between the upper and lower functions, like $y_{top} - y_{bottom}$ .

Volume Integral

Criticality: 3

The definite integral $V = \int_a^b A(x) dx$ (or $V = \int_c^d A(y) dy$), used to calculate the total volume of a solid by summing the infinitesimal volumes of its cross-sections.

Example:

To find the volume of a solid with known cross-sections, you would set up and evaluate a Volume Integral like $\int_0^2 (x^2) dx$ if the cross-sections were squares with side length $x$ .

Volumes with Cross Sections

Criticality: 3

A method to calculate the volume of a 3D solid by integrating the area of its 2D cross-sections taken perpendicular to an axis over a given interval.

Example:

To find the volume of a solid whose base is a circle and whose cross-sections perpendicular to the x-axis are squares, you would use the Volumes with Cross Sections method.