Volume: Measurement, Solid Geometry, and Integral Methods

All flashcards

Capacity

The maximum volume a container can hold.

Cylinder volume

For radius $r$ and perpendicular height $h$, $V=\pi r^2h$.

Volume mesh

A model representing an object's interior volume and relevant surface properties.

Volumetric flow rate

The volume of fluid passing through a surface per unit time.

Quiz questions

1. What does the capacity of a container describe?Answer: The maximum volume the container can hold.
2. Which equation relates mass $m$, density $\rho$, and volume $V$?Answer: $m=\rho V$
3. A cuboid measures $4\,\text{cm}$ by $3\,\text{cm}$ by $2\,\text{cm}$. What is its volume?Answer: $24\,\text{cm}^3$
4. In cylindrical coordinates $(r,\theta,z)$, which expression gives the volume element?Answer: $r\,dr\,d\theta\,dz$
5. A plane region between radii $R(x)$ and $r(x)$ is revolved about the $x$-axis. Which expression gives its washer-method volume?Answer: $\pi\int_a^b\left(R(x)^2-r(x)^2\right)\,dx$