volume

Examples of volume in the following topics:

• Volumes

  • Three dimensional mathematical shapes are also assigned volumes.
  • A volume integral is a triple integral of the constant function $1$, which gives the volume of the region $D$.
  • Using the triple integral given above, the volume is equal to:
  • Triple integral of a constant function $1$ over the shaded region gives the volume.
  • Calculate the volume of a shape by using the triple integral of the constant function 1

• Cylindrical Shells

  • Integration, as an accumulative process, can then calculate the integrated volume of a "family" of shells (a shell being the outer edge of a hollow cylinder), giving us the total volume.
  • Therefore, the entire integrand, $2\pi x \left | f(x) - g(x) \right | \,dx$, is nothing but the volume of the cylindrical shell.
  • By adding the volumes of all these infinitely thin cylinders, we can calculate the volume of the solid formed by the revolution.
  • The volume of solid formed by rotating the area between the curves of $f(y)$ and and the lines $y=a$ and $y=b$ about the $x$-axis is given by:
  • Calculating volume using the shell method.

• Volumes of Revolution

  • Disc and shell methods of integration can be used to find the volume of a solid produced by revolution.
  • Here, we will study how to compute volumes of these objects.
  • Summing up all of the areas along the interval gives the total volume.
  • The volume of each infinitesimal disc is therefore:
  • Summing up all of the surface areas along the interval gives the total volume.

• Optimization in Several Variables

  • A packaging company needs cardboard boxes in rectangular cuboid shape with a given volume of 1000 cubic centimeters and would like to minimize the material cost for the boxes.
  • The constraint in the case is that the volume is fixed: $V = xyz = 1000$.
  • That is to say, the box that minimizes the cost of materials while maintaining the desired volume should be a 10-by-10-by-10 cube.
  • Mathematical optimization can be used to solve problems that involve finding the right size of a volume such as a cuboid.

• Center of Mass and Inertia

  • If the mass distribution is continuous with the density $\rho (r)$ within a volume $V$, then the integral of the weighted position coordinates of the points in this volume relative to the center of mass $\mathbf{R}$ is zero; that is:
  • where $M$ is the total mass in the volume.
  • The integral is over the three dimensional volume, so it is a triple integral.

• The Divergence Theorem

  • More precisely, the divergence theorem states that the outward flux of a vector field through a closed surface is equal to the volume integral of the divergence over the region inside the surface.
  • Suppose $V$ is a subset of $R^n$ (in the case of $n=3$, $V$ represents a volume in 3D space) which is compact and has a piecewise smooth boundary $S$ (also indicated with $\partial V=S$).
  • The left side is a volume integral over the volume $V$; the right side is the surface integral over the boundary of the volume $V$.
  • The divergence theorem can be used to calculate a flux through a closed surface that fully encloses a volume, like any of the surfaces on the left.

• Physics and Engeineering: Center of Mass

  • If the mass distribution is continuous with respect to the density, $\rho (r)$, within a volume, $V$, then the integral of the weighted position coordinates of the points in this volume relative to the center of mass, $\mathbf{R}$, is zero, that is:
  • where $M$ is the total mass in the volume.
  • If a continuous mass distribution has uniform density, which means $\rho$ is constant, then the center of mass is the same as the centroid of the volume.

• Double Integrals Over Rectangles

  • Just as the definite integral of a positive function of one variable represents the area of the region between the graph of the function and the $x$-axis, the double integral of a positive function of two variables represents the volume of the region between the surface defined by the function (on the three dimensional Cartesian plane where $z = f(x, y))$ and the plane which contains its domain.
  • The same volume can be obtained via the triple integral—the integral of a function in three variables—of the constant function $f(x, y, z) = 1$ over the above-mentioned region between the surface and the plane.
  • Double integral as volume under a surface $z = x^2 − y^2$.
  • Use double integrals to find the volume of rectangular regions in the xy-plane

• Applications of Multiple Integrals

  • In mechanics, the moment of inertia is calculated as the volume integral (triple integral) of the density weighed with the square of the distance from the axis:
  • If there is a continuous function $\rho(x)$ representing the density of the distribution at $x$, so that $dm(x) = \rho (x)d^3x$, where $d^3x$ is the Euclidean volume element, then the gravitational potential is:
  • In the following example, the electric field produced by a distribution of charges given by the volume charge density $\rho (\vec r)$ is obtained by a triple integral of a vector function:

• Related Rates

  • The volume is changing at a rate of 2 cubic feet per minute.
  • The relevant formulas and pieces of information are the volume of the balloon, the rate of change of the volume, and the radius.