## Triple Integration-Integration Formulas

### Triple Integration

Functions of three variables: f(x,y,z), g(x,y,z), f(u,v,w)

Independent variables: x, y, z, u, v, w

Small changes: Δxi, Δyj, Δzk

Limits of integration: a, b, c, d, r, s

Domains of integration: G, T, S

Cylindrical coordinates: r, θ, z

Spherical coordinates: r, θ, φ

Volume of a solid: V

Mass of a solid: m

Density: μ(x,y,z)

Coordinates of the center of mass: xc, yc, zc

First moments: Mxy, Myz, Mxz

Moments of inertia: Ixy, Iyz, Ixz, Ix, Iy, Iz, I0

### 1. The triple integral of a function f(x,y,z) over a parallelepiped [a,b]×[c,d]×[r,s] is defined to be the limit of the integral sum (Riemann sum):

$$\iiint \limits_{[a,b]×[c,d]×[r,s]} f(x,y,z) \ dV= \lim_{\substack{\text{max} \ \Delta {x_i} \to 0 \\ \text{max} \ \Delta {y_j} \to 0 \\ \text{max} \ \Delta {z_k} \to 0 }} \sum_{i=1}^m \sum_{j=1}^n \sum_{k=1}^p f(u_i,v_j,w_k) \Delta x_i \Delta y_j \Delta z_k,$$

where (ui,vj,wk) is some point in the parallelepiped (xi−1,xi)×(yj−1,yj)×(zk−1,zk), and the corresponding increments of the variables are equal to Δxi=xi−xi−1, Δyj=yj−yj−1, Δzk=zk−zk−1.

### 2. The triple integral of the sum of two functions is equal to the sum of the integrals of these functions:

$$\iiint \limits_G [f(x,y,z)+g(x,y,z)] \ dV = \iiint \limits_G f(x,y,z) \ dV + \iiint \limits_G g(x,y,z) \ dV$$

### 3. The triple integral of the difference of two functions is equal to the difference of the corresponding integrals of these functions:

$$\iiint \limits_G [f(x,y,z)-g(x,y,z)] \ dV = \iiint \limits_G f(x,y,z) \ dV – \iiint \limits_G g(x,y,z) \ dV$$

### 4. A constant factor can be moved across the triple integral sign:

$$\iiint \limits_G kf(x,y,z) \ dV = k \iiint \limits_G f(x,y,z) \ dV$$

### 5. If f(x,y,z) ≥ 0 and G and T are non-overlapping regions, then

$$\iiint \limits_{G\cup T} f(x,y,z) \ dV = \iiint \limits_G f(x,y,z) \ dV + \iiint \limits_T f(x,y,z) \ dV$$

Here G∪T is the union of the regions of integrations G and

### 6. Expressing a triple integral as a double integral If the integration domain G consists of a set of points (x,y,z) satisfying the condition

(x,y) ∈ R, λ1(x,y) ≤ z ≤ λ2(x,y), then the triple integral is expressed as T.

$$\iiint \limits_G f(x,y,z) dxdydz= \iint \limits_R \left[ \int\limits_{\lambda_1 (x,y)}^{\lambda_2 (x,y)} f(x,y,z) dz \right] dxdy$$

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 (x,y,z) such that

a ≤ x ≤ b, φ1(x) ≤ y ≤ φ2(x), λ1(x,y) ≤ z ≤ λ2(x,y), then the triple integral is given by

$$\iiint \limits_G f(x,y,z) \ dxdydz = \int\limits_a^b \left[ \int\limits_{\varphi_1(x)}^{\varphi_2(x)} \left( \int\limits_{\lambda_1 (x,y)}^{\lambda_2 (x,y)} f(x,y,z) dz \right) \ dy \right] dx$$

### 8. Triple integral over a parallelepiped

If the domain of integration G is a parallelepiped [a,b]×[c,d] ×[r,s], then

$$\iiint \limits_G f(x,y,z) \ dxdydz = \int\limits_a^b \left[ \int\limits_c^d \left( \int\limits_r^s f(x,y,z) dz \right) \ dy \right] dx$$

In the special case when the integrand f(x,y,z) can be written as the product g(x)h(y)k(z), the triple integral is given by

$$\iiint \limits_G f(x,y,z) \ dxdydz = \left( \int\limits_a^b g(x) dx \right) \left( \int\limits_c^d h(y) dy \right) \left( \int\limits_T^s k(z) dz \right)$$

### 9. Change of variables

$$\iint \limits_G f(x,y,z) \ dxdydz = \iint \limits_S f[x(u,v,w),y(u,v,w),z(u,v,w)] \left| \frac {\partial (x,y,z)}{\partial (u,v,w)} \right| dxdydz$$
$$where, \ \left| \frac {\partial (x,y,z)}{\partial (u,v,w)} \right| = \left\vert \matrix{ \frac {\partial x}{\partial u} & \frac {\partial x}{\partial v} & \frac {\partial x}{\partial w} \cr \frac {\partial y}{\partial u} & \frac {\partial y}{\partial v} & \frac {\partial y}{\partial w} \cr \frac {\partial z}{\partial u} & \frac {\partial z}{\partial v} & \frac {\partial z}{\partial w} } \right\vert \neq 0$$

is the jacobian of the transformation (x,y,z)→(u,v,w) and S is the pullback of the integration domain G, which can be computed by substituting x=x(u,v,w), y=y(u,v,w), z=z(u,v,w) into the definition of G.

### 10. Triple integral in cylindrical coordinates

The differential dxdydz in cylindrical coordinates is defined by the expression

$$dxdydz= \left| \frac {\partial (x,y,z)}{\partial (r,\theta,z)}\right| drd\theta dz = rdrd\theta dz$$

Let the solid G is determined by the inequalities

(x,y) ∈ R, λ1(x,y) ≤ z ≤ λ2(x,y), where R is the projection of G onto the xy-plane. Then

$$\iiint \limits_G f(x,y,z) \ dxdydz = \iiint \limits_S f(r \cos \theta , r \sin \theta , z) rdrd \theta dz = \iiint \limits_{R(r,\theta ) } \left[ \int\limits_{\lambda_1 (r \cos \theta , r \sin \theta}^{\lambda_2 (r \cos \theta , r \sin \theta} f(r \cos \theta, r \sin \theta , z)dz \right] rdrd\theta$$

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

$$dxdydz= \left| \frac {\partial (x,y,z)}{\partial (r,\theta,\varphi)}\right| drd\theta d\varphi = r^2 \sin \theta drd\theta d\varphi$$

In spherical coordinates, the triple integral is written as

$$\iiint \limits_G f(x,y,z) \ dxdydz = \iiint \limits_S f(r \sin \theta \cos \varphi , r \sin \theta \sin \varphi, r \cos \theta) r^2 \sin \theta drd \theta d\varphi$$

where S is the pullback of G in spherical coordinates. The angle θ ranges from 0 to 2π, the angle φ ranges from 0 to π.

### 12. Volume of a solid

$$V= \iiint \limits_G dxdydz$$

### 13. Volume of a solid in cylindrical coordinates

$$V= \iiint \limits_S(r, \theta , z) rdr d \theta dz$$

### 14. Volume of a solid in spherical coordinates

$$V= \iiint \limits_S(r, \theta , \varphi) r^2 \sin \theta dr d \theta d \varphi$$

### 15. Mass of a solid

$$m= \iiint \limits_G \mu (x,y,z) \ dV$$

where the solid occupies the domain G and its density at a point (x,y,z) is equal to μ(x,y,z).

### 16. Center of mass of a solid

$$x_c= \frac {M_{yz}}{m}, y_c= \frac {M_{xz}}{m}, z_c= \frac {M_{xy}}{m}, \ where$$
$$M_{yz}=\iiint \limits_G x \mu (x,y,z) \ dV$$
$$M_{xz}=\iiint \limits_G y \mu (x,y,z) \ dV$$
$$M_{xy}=\iiint \limits_G z \mu (x,y,z) \ dV$$

are the first moments about the coordinate planes x=0, y=0 and z=0, respectively, and the function μ(x,y,z) describes the density of the solid.

### 17. Moments of inertia about the xy-plane (or z=0), yz-plane (x=0), and xz-plane (y=0)

$$I_{xy}=\iiint \limits_G z^2 \mu (x,y,z) \ dV$$
$$I_{yz}=\iiint \limits_G x^2 \mu (x,y,z) \ dV$$
$$I_{xz}=\iiint \limits_G y^2 \mu (x,y,z) \ dV$$

### 18. Moments of inertia about the x-axis, y-axis, and z-axis

$$I_x=I_{xy}+I_{xz}= \iiint \limits_G (x^2+y^2) \mu (x,y,z) \ dV$$
$$I_y=I_{xy}+I_{yz}= \iiint \limits_G (z^2+x^2) \mu (x,y,z) \ dV$$
$$I_z=I_{yz}+I_{xz}= \iiint \limits_G (y^2+x^2) \mu (x,y,z) \ dV$$

### 19.Polar moment of inertia

$$I_0=I_{xy}+I_{yz}+I_{xz}= \iiint \limits_G (x^2+y^2+z^2) \mu (x,y,z) \ dV$$