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: \(r,\) \(\theta,\) \(\varphi\)
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: \({I_{xy}},\) \({I_{yz}},\) \({I_{xz}},\) \({I_x},\) \({I_y},\) \({I_z},\) \({I_0}\)
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: \({I_{xy}},\) \({I_{yz}},\) \({I_{xz}},\) \({I_x},\) \({I_y},\) \({I_z},\) \({I_0}\)
- 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 \(\Delta {x_i} = {x_i} – {x_{i – 1}},\) \(\Delta {y_j} = {y_j} – {y_{j – 1}},\) \(\Delta {z_k} = {z_k} – {z_{k – 1}}.\) - The triple integral of the sum of two functions is equal to the sum of the integrals of these functions:
\({\large\iiint\limits_G\normalsize} {\left[ {f\left( {x,y,z} \right) + g\left( {x,y,z} \right)} \right]dV} =\) \({\large\iiint\limits_G\normalsize} {f\left( {x,y,z} \right)dV} \) \(+\;{\large\iiint\limits_G\normalsize} {g\left( {x,y,z} \right)dV} \) - The triple integral of the difference of two functions is equal to the difference of the corresponding integrals of these functions:
\({\large\iiint\limits_G\normalsize} {\left[ {f\left( {x,y,z} \right) – g\left( {x,y,z} \right)} \right]dV} =\) \({\large\iiint\limits_G\normalsize} {f\left( {x,y,z} \right)dV} \) \(-\;{\large\iiint\limits_G\normalsize} {g\left( {x,y,z} \right)dV} \) - A constant factor can be moved across the triple integral sign:
\({\large\iiint\limits_G\normalsize} {kf\left( {x,y,z} \right)dV} =\) \(k{\large\iiint\limits_G\normalsize} {f\left( {x,y,z} \right)dV} \) - 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 \(T\). - 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. - 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
\({\large\iiint\limits_G\normalsize} {f\left( {x,y,z} \right)dxdydz} =\) \({\large\int\limits_a^b\normalsize} {\left[ {{\large\int\limits_{{\varphi _1}\left( x \right)}^{{\varphi _2}\left( x \right)}\normalsize} }\right.}\) \({\left.{ {\left( {{{\large\int\limits_{{\lambda _1}\left( {x,y} \right)}^{{\lambda _2}\left( {x,y} \right)}\normalsize}} {f\left( {x,y,z} \right)dz} } \right) dy} } \right]dx} \) - 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
\({\large\iiint\limits_G\normalsize} {f\left( {x,y,z} \right)dxdydz} =\) \( \left( {{\large\int\limits_a^b\normalsize} {g\left( x \right)dx} } \right)\) \(\left( {{\large\int\limits_c^d\normalsize} {h\left( y \right)dy} } \right)\) \(\left( {{\large\int\limits_r^s\normalsize} {k\left( z \right)dz} } \right)\) - 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 \(G.\) - 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. - 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 \(\pi.\) - Volume of a solid
\(V = {\large\iiint\limits_G\normalsize} {dxdydz} \) - Volume of a solid in cylindrical coordinates
\(V = {\large\iiint\limits_{S\left( {r,\theta ,z} \right)}\normalsize} {rdrd\theta dz} \) - Volume of a solid in spherical coordinates
\(V = {\large\iiint\limits_{S\left( {r,\theta ,\varphi } \right)}\normalsize} {{r^2}\sin \theta drd\theta d\varphi } \) - 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 \({\mu \left( {x,y,z} \right)}.\) - 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. - Moments of inertia about the \(xy\)-plane (or \(z = 0\text{),}\) \(yz\)-plane \(\left({x = 0}\right),\) and \(xz\)-plane \(\left({y = 0}\right)\)
\({I_{xy}} = {\large\iiint\limits_G\normalsize} {{z^2}\mu \left( {x,y,z} \right)dV} ,\)
\({I_{yz}} = {\large\iiint\limits_G\normalsize} {{x^2}\mu \left( {x,y,z} \right)dV} ,\)
\({I_{xz}} = {\large\iiint\limits_G\normalsize} {{y^2}\mu \left( {x,y,z} \right)dV} \) - Moments of inertia about the \(x\)-axis, \(y\)-axis, and \(z\)-axis
\({I_x} = {I_{xy}} + {I_{xz}} =\) \({\large\iiint\limits_G\normalsize} {\left( {{z^2} + {y^2}} \right)\mu \left( {x,y,z} \right)dV}, \)
\({I_y} = {I_{xy}} + {I_{yz}} =\) \({\large\iiint\limits_G\normalsize} {\left( {{z^2} + {x^2}} \right)\mu \left( {x,y,z} \right)dV}, \)
\({I_z} = {I_{xz}} + {I_{yz}} =\) \({\large\iiint\limits_G\normalsize} {\left( {{y^2} + {x^2}} \right)\mu \left( {x,y,z} \right)dV} \) - Polar moment of inertia
\({I_0} = {I_{xy}} + {I_{yz}} + {I_{xz}} =\) \({\large\iiint\limits_G\normalsize} {\left( {{x^2} + {y^2} + {z^2}} \right) }\) \({\mu \left( {x,y,z} \right)dV} \)
