Pappus, Composite and Advanced Solids — Worked Examples

An ellipsoidal tank has semi-axes $a=4.00\,\text{m}$, $b=3.00\,\text{m}$, and $c=2.00\,\text{m}$. Determine its geometric volume.

A ring torus has major radius $R=5.00\,\text{m}$ and tube radius $r=1.20\,\text{m}$. Determine its volume and surface area.

A circle of radius $1.20\,\text{m}$ is revolved about an external coplanar axis $5.00\,\text{m}$ from the circle center. Use Pappus-Guldinus to determine the generated volume.

Use Pappus-Guldinus to determine the surface area generated when a circle of radius $1.20\,\text{m}$ is revolved about the same external axis $5.00\,\text{m}$ from its center.

A semicircular area of radius $3.00\,\text{m}$ is revolved through $360^\circ$ about its bounding diameter. Use Pappus-Guldinus to determine the generated volume.

A semicircular wire arc of radius $r=2.00\,\text{m}$ is revolved about a coplanar line parallel to its diameter and located $5.00\,\text{m}$ from the diameter on the side opposite the arc. Determine the generated surface area.

A square frustum has end areas $A_1=100\,\text{m}^2$ and $A_2=36.0\,\text{m}^2$. The corresponding side lengths are therefore $10.0\,\text{m}$ and $6.00\,\text{m}$. Its height is $9.00\,\text{m}$. Determine the volume using the prismatoidal formula.

Five equally spaced cross-sections are $10.0\,\text{m}$ apart and have areas $20.0$, $28.0$, $35.0$, $41.0$, and $44.0\,\text{m}^2$. Estimate the enclosed volume using the composite trapezoidal rule.

Use the five equally spaced areas from Example 8 to estimate volume with Simpson's $1/3$ rule.

Four measured cross-sections occur at stations $0$, $8$, $18$, and $30\,\text{m}$. Their areas are known, but the spacing is not uniform. Determine whether the equal-spacing composite Simpson's $1/3$ formula from the lesson may be applied directly.

A tank consists of a vertical cylinder of radius $2.00\,\text{m}$ and height $5.00\,\text{m}$ topped by a hemisphere of the same radius. Determine its total internal geometric capacity.

A rectangular concrete block measures $4.00\,\text{m}\times3.00\,\text{m}\times2.00\,\text{m}$. A cylindrical opening of radius $0.500\,\text{m}$ is bored completely through the $2.00\,\text{m}$ dimension. Determine the remaining concrete volume.