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Calculus

# Properties and Applications of Triple Integrals

Functions of three variables: $$f\left( {x,y,z} \right),$$ $$g\left( {x,y,z} \right),$$ $$f\left( {u,v,w} \right)$$
Independent variables: $$x,$$ $$y,$$ $$z,$$ $$u,$$ $$v,$$ $$w$$
Small changes: $$\Delta {x_i},$$ $$\Delta {y_j},$$ $$\Delta {z_k}$$
Limits of integration: $$a,$$ $$b,$$ $$c,$$ $$d,$$ $$r,$$ $$s$$
Domains of integration: $$G,$$ $$T,$$ $$S$$
Cylindrical coordinates: $$r,$$ $$\theta,$$ $$z$$
Spherical coordinates: $$r,$$ $$\theta,$$ $$\varphi$$

Volume of a solid: $$V$$
Mass of a solid: $$m$$
Density: $$\mu \left( {x,y,z} \right)$$
Coordinates of the center of mass: $${x_c},$$ $${y_c},$$ $${z_c}$$
First moments: $${M_{xy}},$$ $${M_{yz}},$$ $${M_{xz}}$$
Moments of inertia: $${I_{xy}},$$ $${I_{yz}},$$ $${I_{xz}},$$ $${I_x},$$ $${I_y},$$ $${I_z},$$ $${I_0}$$

1. The triple integral of a function $$f\left( {x,y,z} \right)$$ over a parallelepiped $$\left[ {a,b} \right] \times \left[ {c,d} \right] \times \left[ {r,s} \right]$$ is defined to be the limit of the integral sum (Riemann sum):
$$\require{AMSmath.js}{\large\iiint\limits_{\left[ {a,b} \right] \times \left[ {c,d} \right] \times \left[ {r,s} \right]}\normalsize} {f\left( {x,y,z} \right)dV} =$$ $$\lim\limits_{\substack{ \text{max}\,\Delta {x_i} \to 0\\ \text{max}\,\Delta {y_j} \to 0\\ \text{max}\,\Delta {z_k} \to 0}} \sum\limits_{i = 1}^m {\sum\limits_{j = 1}^n {\sum\limits_{k = 1}^p {f\left( {{u_i},{v_j},{w_k}} \right)}}}$$ $${{{ \Delta {x_i}\Delta {y_j}\Delta {z_k}} } },$$
where $${\left( {{u_i},{v_j},{w_k}} \right)}$$ is some point in the parallelepiped $$\left( {{x_{i – 1}},{x_i}} \right) \times \left( {{y_{j – 1}},{y_j}} \right)$$ $$\times\, \left( {{z_{k – 1}},{z_k}} \right),$$ and the corresponding increments of the variables are equal to $$\Delta {x_i} = {x_i} – {x_{i – 1}},$$ $$\Delta {y_j} = {y_j} – {y_{j – 1}},$$ $$\Delta {z_k} = {z_k} – {z_{k – 1}}.$$
2. The triple integral of the sum of two functions is equal to the sum of the integrals of these functions:
$${\large\iiint\limits_G\normalsize} {\left[ {f\left( {x,y,z} \right) + g\left( {x,y,z} \right)} \right]dV} =$$ $${\large\iiint\limits_G\normalsize} {f\left( {x,y,z} \right)dV}$$ $$+\;{\large\iiint\limits_G\normalsize} {g\left( {x,y,z} \right)dV}$$
3. The triple integral of the difference of two functions is equal to the difference of the corresponding integrals of these functions:
$${\large\iiint\limits_G\normalsize} {\left[ {f\left( {x,y,z} \right) – g\left( {x,y,z} \right)} \right]dV} =$$ $${\large\iiint\limits_G\normalsize} {f\left( {x,y,z} \right)dV}$$ $$-\;{\large\iiint\limits_G\normalsize} {g\left( {x,y,z} \right)dV}$$
4. A constant factor can be moved across the triple integral sign:
$${\large\iiint\limits_G\normalsize} {kf\left( {x,y,z} \right)dV} =$$ $$k{\large\iiint\limits_G\normalsize} {f\left( {x,y,z} \right)dV}$$
5. If $${f\left( {x,y,z} \right)} \ge 0$$ and $$G$$ and $$T$$ are non-overlapping regions, then
$${\large\iiint\limits_{G \cup T}\normalsize} {f\left( {x,y,z} \right)dV} =$$ $${\large\iiint\limits_G\normalsize} {f\left( {x,y,z} \right)dV}$$ $$+\;{\large\iiint\limits_T\normalsize} {f\left( {x,y,z} \right)dV}$$
Here $$G \cup T$$ is the union of the regions of integrations $$G$$ and $$T$$.
6. Expressing a triple integral as a double integral
If the integration domain $$G$$ consists of a set of points $${\left( {x,y,z} \right)}$$ satisfying the condition
$$\left( {x,y} \right) \in \mathbb{R},$$ $${\lambda _1}\left( {x,y} \right) \le z \le {\lambda _2}\left( {x,y} \right),$$
then the triple integral is expressed as
$${\large\iiint\limits_G\normalsize} {f\left( {x,y,z} \right)dxdydz} =$$ $${\large\iint\limits_R\normalsize} {\left[ {{\large\int\limits_{{\lambda _1}\left( {x,y} \right)}^{{\lambda _2}\left( {x,y} \right)}\normalsize} {f\left( {x,y,z} \right)dz} } \right]dxdy} ,$$
where $$R$$ is the projection of $$G$$ onto the $$xy$$-plane.
7. Expressing a triple integral as an iterated integral
If the integration domain $$G$$ consists of a set of points $${\left( {x,y,z} \right)}$$ such that
$$a \le x \le b,$$ $${\varphi _1}\left( x \right) \le y \le {\varphi _2}\left( x \right),$$ $${\lambda _1}\left( {x,y} \right) \le z \le {\lambda _2}\left( {x,y} \right),$$
then the triple integral is given by
$${\large\iiint\limits_G\normalsize} {f\left( {x,y,z} \right)dxdydz} =$$ $${\large\int\limits_a^b\normalsize} {\left[ {{\large\int\limits_{{\varphi _1}\left( x \right)}^{{\varphi _2}\left( x \right)}\normalsize} }\right.}$$ $${\left.{ {\left( {{{\large\int\limits_{{\lambda _1}\left( {x,y} \right)}^{{\lambda _2}\left( {x,y} \right)}\normalsize}} {f\left( {x,y,z} \right)dz} } \right) dy} } \right]dx}$$
8. Triple integral over a parallelepiped
If the domain of integration $$G$$ is a parallelepiped $$\left[ {a,b} \right] \times \left[ {c,d} \right]$$ $$\times\, \left[ {r,s} \right]$$, then
$${\large\iiint\limits_G\normalsize} {f\left( {x,y,z} \right)dxdydz} =$$ $${\large\int\limits_a^b\normalsize} {\left[ {{\large\int\limits_c^d\normalsize} {\left( {{\large\int\limits_r^s\normalsize} {f\left( {x,y,z} \right)dz} } \right)dy} } \right]dx}$$
In the special case when the integrand $${f\left( {x,y,z} \right)}$$ can be written as the product
$$g\left( x \right)h\left( y \right)k\left( z \right),$$ the triple integral is given by
$${\large\iiint\limits_G\normalsize} {f\left( {x,y,z} \right)dxdydz} =$$ $$\left( {{\large\int\limits_a^b\normalsize} {g\left( x \right)dx} } \right)$$ $$\left( {{\large\int\limits_c^d\normalsize} {h\left( y \right)dy} } \right)$$ $$\left( {{\large\int\limits_r^s\normalsize} {k\left( z \right)dz} } \right)$$
9. Change of variables
$${\large\iiint\limits_G\normalsize} {f\left( {x,y,z} \right)dxdydz} =$$ $${\large\iiint\limits_S\normalsize} {f\big[ {x\left( {u,v,w} \right),y\left( {u,v,w} \right), }}$$ $${{ z\left( {u,v,w} \right)} \big]\left|{\large\frac{{\partial \left( {x,y,z} \right)}}{{\partial \left( {u,v,w} \right)}}\normalsize}\right| dxdydz},$$ where $$\left| {{\large\frac{{\partial \left( {x,y,z} \right)}}{{\partial \left( {u,v,w} \right)}}}\normalsize} \right| =$$ $$\left| {\begin{array}{*{20}{c}} {\large\frac{{\partial x}}{{\partial u}}\normalsize} & {\large\frac{{\partial x}}{{\partial v}}\normalsize} & {\large\frac{{\partial x}}{{\partial w}}\normalsize}\\ {\large\frac{{\partial y}}{{\partial u}}\normalsize} & {\large\frac{{\partial y}}{{\partial v}}\normalsize} & {\large\frac{{\partial y}}{{\partial w}}\normalsize}\\ {\large\frac{{\partial z}}{{\partial u}}\normalsize} & {\large\frac{{\partial z}}{{\partial v}}\normalsize} & {\large\frac{{\partial z}}{{\partial w}}\normalsize} \end{array}} \right|$$ $$\ne 0$$ is the jacobian of the transformation
$$\left( {x,y,z} \right) \to \left( {u,v,w} \right)$$ and $$S$$ is the pullback of the integration domain $$G,$$ which can be computed by substituting $$x = x\left( {u,v,w} \right),$$ $$y = y\left( {u,v,w} \right),$$ $$z = z\left( {u,v,w} \right)$$ into the definition of $$G.$$
10. Triple integral in cylindrical coordinates
The differential $$dxdydz$$ in cylindrical coordinates is defined by the expression
$$dxdydz = \left|{\large\frac{{\partial \left( {x,y,z} \right)}}{{\partial \left( {r,\theta ,z} \right)}}\normalsize}\right| drd\theta dz =$$ $$rdrd\theta dz.$$
Let the solid $$G$$ is determined by the inequalities
$$\left( {x,y} \right) \in \mathbb{R},$$ $${\lambda _1}\left( {x,y} \right) \le z \le {\lambda _2}\left( {x,y} \right),$$
where $$R$$ is the projection of $$G$$ onto the $$xy$$-plane. Then
$${\large\iiint\limits_G\normalsize} {f\left( {x,y,z} \right)dxdydz} =$$ $${\large\iiint\limits_S\normalsize} {f\left( {r\cos \theta ,r\sin \theta ,z} \right)rdrd\theta dz} =$$ $${\large\iint\limits_{R\left( {r,\theta } \right)}\normalsize} {\Big[ {{\large\int\limits_{{\lambda _1}\left( {r\cos \theta ,r\sin \theta } \right)}^{{\lambda _2}\left( {r\cos \theta ,r\sin \theta } \right)}\normalsize} {f\big( {r\cos \theta , }}}}$$ $${{{{ r\sin \theta ,z} \big)dz} } \Big]rdrd\theta }.$$
Here $$S$$ is the pullback of $$G$$ in cylindrical coordinates.
11. Triple integral in spherical coordinates
The differential $$dxdydz$$ in spherical coordinates is expressed by the formula
$$dxdydz = \left| {{\large\frac{{\partial \left( {x,y,z} \right)}}{{\partial \left( {r,\theta ,\varphi } \right)}}}\normalsize} \right|drd\theta d\varphi =$$ $${r^2}\sin \theta drd\theta d\varphi$$
In spherical coordinates, the triple integral is written as
$${\large\iiint\limits_G\normalsize} {f\left( {x,y,z} \right)dxdydz} =$$ $${\large\iiint\limits_S\normalsize} {f\big( {r\sin \theta \cos \varphi ,r\sin \theta \sin \varphi , }}$$ $${{ r\cos \theta } \big){r^2}\sin \theta drd\theta d\varphi },$$
where $$S$$ is the pullback of $$G$$ in spherical coordinates. The angle $$\theta$$ ranges from $$0$$ to $$2\pi$$, the angle $$\varphi$$ ranges from $$0$$ to $$\pi.$$
1. Volume of a solid
$$V = {\large\iiint\limits_G\normalsize} {dxdydz}$$
2. Volume of a solid in cylindrical coordinates
$$V = {\large\iiint\limits_{S\left( {r,\theta ,z} \right)}\normalsize} {rdrd\theta dz}$$
3. Volume of a solid in spherical coordinates
$$V = {\large\iiint\limits_{S\left( {r,\theta ,\varphi } \right)}\normalsize} {{r^2}\sin \theta drd\theta d\varphi }$$
4. Mass of a solid
$$m = {\large\iiint\limits_G\normalsize} {\mu \left( {x,y,z} \right)dV} ,$$
where the solid occupies the domain $$G$$ and its density at a point $${\left( {x,y,z} \right)}$$ is equal to $${\mu \left( {x,y,z} \right)}.$$
5. Center of mass of a solid
$${x_C} = {\large\frac{{{M_{yz}}}}{m}\normalsize},\;$$ $${y_C} = {\large\frac{{{M_{xz}}}}{m}\normalsize},\;$$ $${z_C} = {\large\frac{{{M_{xy}}}}{m}\normalsize}$$, where
$${M_{yz}} = {\large\iiint\limits_G\normalsize} {x\mu \left( {x,y,z} \right)dV},\;$$
$${M_{xz}} = {\large\iiint\limits_G\normalsize} {y\mu \left( {x,y,z} \right)dV},\;$$
$${M_{xy}} = {\large\iiint\limits_G\normalsize} {z\mu \left( {x,y,z} \right)dV}$$
are the first moments about the coordinate planes $$x = 0,$$ $$y = 0$$ and $$z = 0,$$ respectively, and the function $${\mu \left( {x,y,z} \right)}$$ describes the density of the solid.
6. Moments of inertia about the $$xy$$-plane (or $$z = 0\text{),}$$ $$yz$$-plane $$\left({x = 0}\right),$$ and $$xz$$-plane $$\left({y = 0}\right)$$
$${I_{xy}} = {\large\iiint\limits_G\normalsize} {{z^2}\mu \left( {x,y,z} \right)dV} ,$$
$${I_{yz}} = {\large\iiint\limits_G\normalsize} {{x^2}\mu \left( {x,y,z} \right)dV} ,$$
$${I_{xz}} = {\large\iiint\limits_G\normalsize} {{y^2}\mu \left( {x,y,z} \right)dV}$$
7. Moments of inertia about the $$x$$-axis, $$y$$-axis, and $$z$$-axis
$${I_x} = {I_{xy}} + {I_{xz}} =$$ $${\large\iiint\limits_G\normalsize} {\left( {{z^2} + {y^2}} \right)\mu \left( {x,y,z} \right)dV},$$
$${I_y} = {I_{xy}} + {I_{yz}} =$$ $${\large\iiint\limits_G\normalsize} {\left( {{z^2} + {x^2}} \right)\mu \left( {x,y,z} \right)dV},$$
$${I_z} = {I_{xz}} + {I_{yz}} =$$ $${\large\iiint\limits_G\normalsize} {\left( {{y^2} + {x^2}} \right)\mu \left( {x,y,z} \right)dV}$$
8. Polar moment of inertia
$${I_0} = {I_{xy}} + {I_{yz}} + {I_{xz}} =$$ $${\large\iiint\limits_G\normalsize} {\left( {{x^2} + {y^2} + {z^2}} \right) }$$ $${\mu \left( {x,y,z} \right)dV}$$