Chapter 8: Applications of Triple Integration
Section 8.3: First Moments
If R is a three-dimensional region, then its volume V or its total mass m can be computed by one of the integrals in Table 8.3.1. If the density δ in R is 1, the integrals yield the volume of R; otherwise, they yield the total mass in R.
∫∫∫Rδr,θ,z r dv′
∫∫∫δρ,φ,θ ρ2sinφ dv″
Table 8.3.1 Total volume or mass in three-dimensional region R
If the triple integrals in Tables 8.3.1 and 8.3.2 are iterated in Cartesian coordinates, dv is one of the six orderings of the differentials dx, dy, dz; in cylindrical coordinates, dv′ is one of the six orderings of the differentials dr,dz,dθ; and in spherical coordinates, dv″ is one of the six orderings of the differentials dρ,dφ,dθ.
Table 8.3.2 lists the integrals whose values are the first moments for a three-dimensional region R
∫∫∫Rx δ dv
∫∫∫Rr cosθ δ r dv′
∫∫∫Rρ sinφcosθ δ ρ2sinφdv″
∫∫∫Ry δ dv
∫∫∫Rr sinθ δ r dv′
∫∫∫Rρ sinφsinθ δ ρ2sinφdv″
∫∫∫Rz δ dv
∫∫∫Rz δ r dv′
∫∫∫Rρ cosφ δ ρ2sinφdv″
Table 8.3.2 First moments for calculating a centroid (δ=constant) or a center of mass
For a three-dimensional region R, Table 8.3.3 provides the Cartesian coordinates x&conjugate0;,y&conjugate0;,z&conjugate0; of either the centroid or center of mass.
Center of Mass
Table 8.3.3 Centroid or Center of Mass
In each of the following examples, find the centroid of the given region R. Then, find the center of mass under the assumption that the region has the indicated density δ.
R is the region in Example 8.1.3
R is the region in Example 8.1.5
R is the region in Example 8.1.8
R is the region in Example 8.1.14
R is the region in Example 8.1.15
R is the region in Example 8.1.20
R is the region in Example 8.1.21
R is the region in Example 8.1.22
δr ,θ,z=r z cosθ/6
R is the region in Example 8.1.26
δρ,φ,θ=2+ ρ sinφcosθ
R is the region in Example 8.1.27
δx,y,z=2 x2+3 y2+4 z2
R is the region in Example 8.1.28
R is the region in Example 8.1.29
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