Essential Concepts

Triple Integrals

• To compute a triple integral we use Fubini’s theorem, which states that if [latex]f(x,y,z)[/latex] is continuous on a rectangular box
• To compute the volume of a general solid bounded region [latex]E[/latex] we use the triple integral
• Interchanging the order of the iterated integrals does not change the answer. As a matter of fact, interchanging the order of integration can help simplify the computation.
• To compute the average value of a function over a general three-dimensional region, we use

Triple Integrals in Alternative Coordinate Systems

• To evaluate a triple integral in cylindrical coordinates, use the iterated integral
• To evaluate a triple integral in spherical coordinates, use the iterated integral

Center of Mass and Moments of Inertia

• Finding the mass, center of mass, moments, and moments of inertia in double integrals:
  • For a lamina [latex]R[/latex] with a density function [latex]\rho(x,y)[/latex] at any point [latex](x,y)[/latex] in the plane, the mass is [latex]m=\underset{R}{\displaystyle\iint} \rho(x,y) dA[/latex]
  • The moments about the [latex]x[/latex]-axis and [latex]y[/latex]-axis are [latex]M_{x}=\underset{R}{\displaystyle\iint} y\rho(x,y) dA[/latex] and [latex]M_{y}=\underset{R}{\displaystyle\iint} x\rho(x,y) dA[/latex]
  • The center of mass is given by [latex]\overline{x}=\frac{M_{y}}{m}[/latex], [latex]\overline{y}=\frac{M_{x}}{m}[/latex]
  • The center of mass becomes the centroid of the plane when the density is constant.
  • The moments of inertia about the [latex]x[/latex]-axis, [latex]y[/latex]-axis, and the origin are [latex]I_{x}=\displaystyle\iint_{R} y^{2}\rho(x,y) dA[/latex], [latex]I_{y}=\underset{R}{\displaystyle\iint} x^{2}\rho(x,y) dA[/latex], and [latex]I_{0}=I_{x}+I_{y}=\underset{R}{\displaystyle\iint} (x^{2}+y^{2})\rho(x,y) dA[/latex]
• Finding the mass, center of mass, moments, and moments of inertia in triple integrals:
  • For a solid object [latex]Q[/latex] with a density function [latex]\rho(x,y,z)[/latex] at any point [latex](x,y,z)[/latex] in space, the mass is [latex]m=\displaystyle\iiint_{Q} \rho(x,y,z) dV[/latex]
  • The moments about the [latex]xy[/latex]-plane, the [latex]xz[/latex]-plane, and the [latex]yz[/latex]-plane are [latex]M_{xy}=\displaystyle\iiint_{Q} z\rho(x,y,z) dV[/latex], [latex]M_{xz}=\underset{Q}{\displaystyle\iiint} y\rho(x,y,z) dV[/latex], [latex]M_{yz}=\underset{Q}{\displaystyle\iiint} x\rho(x,y,z) dV[/latex]
  • The center of mass is given by [latex]\overline{x}=\frac{M_{yz}}{m}[/latex], [latex]\overline{x}=\frac{M_{xz}}{m}[/latex], [latex]\overline{x}=\frac{M_{xy}}{m}[/latex]
  • The center of mass becomes the centroid of the solid when the density is constant.
  • The moments of inertia about the [latex]yz[/latex]-plane, the [latex]xz[/latex]-plane, and the [latex]xy[/latex]-plane are [latex]I_{x}=\underset{Q}{\displaystyle\iiint} (y^{2}+z^{2})\rho(x,y,z) dV[/latex], [latex]I_{y}=\underset{Q}{\displaystyle\iiint} (x^{2}+z^{2})\rho(x,y,z) dV[/latex], [latex]I_{z}=\underset{Q}{\displaystyle\iiint} (x^{2}+y^{2})\rho(x,y,z) dV[/latex]

Change of Variables

• A transformation [latex]T[/latex] is a function that transforms a region [latex]G[/latex] in one plane (space) into a region [latex]R[/latex] in another plane (space) by a change of variables.
• A transformation [latex]T:G\rightarrow{R}[/latex] defined as [latex]T(u,v)=(x,y)[/latex] [latex](\text{or }T(u,v,w)=(x,y,z))([/latex] is said to be a one-to-one transformation if no two points map to the same image point.
• If [latex]f[/latex] is continuous on [latex]R[/latex], then [latex]\underset{R}{\displaystyle\iint} f(x,y) dA =\underset{S}{\displaystyle\iint} f\left(g(u,v),h(u,v)\right)\left\arrowvert\frac{\partial(x,y)}{\partial(u,v)}\right\arrowvert du dv[/latex]
• If [latex]F[/latex] is continuous on [latex]R[/latex], then [latex]R[/latex], then [latex]\underset{R}{\displaystyle\iiint} F(x,y,z) dV =\underset{G}{\displaystyle\iiint} F\left(g(u,v,w),h(u,v,w),k(u,v,w)\right)\left\arrowvert\frac{\partial(x,y,z)}{\partial(u,v,w)}\right\arrowvert du dv dw = \underset{G}{\displaystyle\iiint}H(u,v,w)|J(u,v,w)| du dv dw[/latex]

Key Equations

• Triple integral
  [latex]\underset{l,m,n\to\infty}{\lim}\displaystyle\sum_{i=1}^{l}\displaystyle\sum_{j=1}^{m}\displaystyle\sum_{k=1}^{n} f(x_{ijk}^{\ast},y_{ijk}^{\ast},z_{ijk}^{\ast}) \Delta{x}\Delta{y}\Delta{z}=\underset{B}{\displaystyle\iiint} f(x,y,z)dV[/latex]
• Triple integral in cylindrical coordinates
  [latex]\underset{B}{\displaystyle\iiint} g(x,y,z)dV=\underset{B}{\displaystyle\iiint} g(r\cos\theta,r\sin\theta,z)r dr d\theta dz=\underset{B}{\displaystyle\iiint} f(r,\theta,z)r dr d\theta dz=[/latex]
• Triple integral in spherical coordinates
  [latex]\underset{B}{\displaystyle\iiint} f(\rho,\theta,\varphi)\rho^{2}\sin\varphi{d}\rho{d}\varphi{d}\theta=\displaystyle\int_{\varphi=\gamma}^{\varphi=\psi}\displaystyle\int_{\theta=\alpha}^{\theta=\beta}\displaystyle\int_{\rho=a}^{\rho=b} \rho^{2}\sin\varphi{d}\rho{d}\varphi{d}\theta[/latex]
• Mass of a lamina
  [latex]m=\underset{k,l\to\infty}{\lim}\displaystyle\sum_{i=1}^{k}\displaystyle\sum_{j=1}^{l}m_{ij}=\underset{k,l\to\infty}{\lim}\displaystyle\sum_{i=1}^{k}\displaystyle\sum_{j=1}^{l}\rho(x_{ij}^{\ast},y_{ij}^{\ast})\Delta{A}=\underset{R}{\displaystyle\iint} \rho(x,y) dA[/latex]
• Moment about the [latex]x[/latex]-axis
  [latex]M_{x}=\underset{k,l\to\infty}{\lim}\displaystyle\sum_{i=1}^{k}\displaystyle\sum_{j=1}^{l}(y_{ij}^{\ast})m_{ij}=\underset{k,l\to\infty}{\lim}\displaystyle\sum_{i=1}^{k}\displaystyle\sum_{j=1}^{l}(y_{ij}^{\ast})\rho(x_{ij}^{\ast},y_{ij}^{\ast})\Delta{A}=\underset{R}{\displaystyle\iint} y\rho(x,y) dA[/latex]
• Moment about the [latex]y[/latex]-axis
  [latex]M_{y}=\underset{k,l\to\infty}{\lim}\displaystyle\sum_{i=1}^{k}\displaystyle\sum_{j=1}^{l}(x_{ij}^{\ast})m_{ij}=\underset{k,l\to\infty}{\lim}\displaystyle\sum_{i=1}^{k}\displaystyle\sum_{j=1}^{l}(x_{ij}^{\ast})\rho(x_{ij}^{\ast},y_{ij}^{\ast})\Delta{A}=\underset{R}{\displaystyle\iint} x\rho(x,y) dA[/latex]
• Center of mass of a lamina
  [latex]\overline{x}=\frac{M_{y}}{m}=\dfrac{\underset{R}{\displaystyle\iint} x\rho(x,y) dA}{\underset{R}{\displaystyle\iint} \rho(x,y) dA}[/latex] and [latex]\overline{y}=\frac{M_{x}}{m}=\dfrac{\underset{R}{\displaystyle\iint} y\rho(x,y) dA}{\underset{R}{\displaystyle\iint} \rho(x,y) dA}[/latex]

Glossary

Jacobian

the Jacobian [latex]J(u ,v)[/latex] in two variables is a [latex]2{\times}2[/latex] determinant:

[latex]J(u,v) = \begin{vmatrix}\frac{dx}{du} & \frac{dy}{du}\\\frac{dx}{dv} & \frac{dy}{dv}\end{vmatrix}[/latex]

the Jacobian [latex]J(u ,v, w)[/latex] in three variables is a [latex]3{\times}3[/latex] determinant:

[latex]J(u,v,w)=\begin{vmatrix}\frac{dx}{du} & \frac{dy}{du} & \frac{dz}{du}\\\frac{dx}{dv} & \frac{dy}{dv} & \frac{dz}{dv}\\\frac{dx}{dw} & \frac{dy}{dw} & \frac{dz}{dw}\end{vmatrix}[/latex]

one-to-one transformation

a transformation [latex]T : G {\rightarrow} R[/latex] defined as [latex]T(u, v) = (x, y)[/latex] is said to be one-to-one if no two points map to the same image point

planar transformation

a function [latex]T[/latex] that transforms a region [latex]G[/latex] in one plane into a region [latex]R[/latex] in another plane by a change of variables

radius of gyration

the distance from an object’s center of mass to its axis of rotation

transformation

a function that transforms a region [latex]G[/latex] in one plane into a region [latex]R[/latex] in another plane by a change of variables

triple integral

the triple integral of a continuous function [latex]f(x, y, z)[/latex]over a rectangular solid box [latex]\bf{B}[/latex] is the limit of a Riemann sum for a function of three variables, if this limit exists

triple integral in cylindrical coordinates

the limit of a triple Riemann sum, provided the following limit exists:[latex]{\displaystyle\lim_{l,m,n\to\infty}{\sum_{i=1}^{l}}{\displaystyle\sum_{j=1}^{m}}{\displaystyle\sum_{k=1}^{n}f({r^{*}}_{i,j,k}, {{\theta}^{*}}_{i,j,k}, {{z}^{*}}_{i,j,k}){r^{*}}_{i,j,k}{\Delta}r{\Delta}{\theta}{\Delta}{z}}}[/latex]

triple integral in spherical coordinates

the limit of a triple Riemann sum, provided the following limit exists: [latex]{\displaystyle\lim_{l,m,n\to\infty}{\displaystyle\sum_{i=1}^{l}}{\displaystyle\sum_{j=1}^{m}}{\displaystyle\sum_{k=1}^{n}f({{\rho}^{*}}_{i,j,k}, {{\theta}^{*}}_{i,j,k}, {{\varphi}^{*}}_{i,j,k})({{\rho}^{*}}_{i,j,k})^{2}\sin{\varphi}{\Delta}{\rho}{\Delta}{\theta}{\Delta}{\varphi}}}[/latex]