Formulas and Tables

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 \(\pi.\)
Triple integral in spherical coordinates
  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
    \({I_0} = {I_{xy}} + {I_{yz}} + {I_{xz}} =\) \({\large\iiint\limits_G\normalsize} {\left( {{x^2} + {y^2} + {z^2}} \right) }\) \({\mu \left( {x,y,z} \right)dV} \)

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