We now explore integration in spherical
coordinates with constant and non-constant limits.
Integral over a Spherical Box
We now want to integrate over the spherical box:
ρ1≤ρ≤ρ2ϕ1≤ϕ≤ϕ2θ1≤θ≤θ2
Using the Riemann Sum definition with
ΔVijk=ρi∗2sinϕj∗ΔρiΔϕjΔθk,
the integral is:
R∭f(ρ,ϕ,θ)dV=p→∞limq→∞limr→∞limi=1∑pj=1∑qk=1∑sf(ρi∗,ϕj∗,θk∗)ρi∗2sinϕj∗ΔρiΔϕjΔθk
We recognize this as the triple iterated integral:
R∭f(ρ,ϕ,θ)dV=∫θ1θ2∫ϕ1ϕ2∫ρ1ρ2f(ρ,ϕ,θ)ρ2sinϕdρdϕdθ
By Fubini's Theorem, there are five other orders
of integrations obtained by reordering the differentials and the
corresponding limits.
Other Integrals in Spherical Coordinates
In addition to integrating over spherical boxes, spherical coordinates can
be used to integrate over other types of regions with appropriate
definitions of the regions. The resulting integrals have the forms:
Type ρθ:R∭f(ρ,ϕ,θ)dV=∫ρ1ρ2∫g(ρ)h(ρ)∫p(ρ,θ)q(ρ,θ)f(ρ,ϕ,θ)ρ2sinϕdϕdθdρType θρ:R∭f(ρ,ϕ,θ)dV=∫θ1θ2∫g(θ)h(θ)∫p(ρ,θ)q(ρ,θ)f(ρ,ϕ,θ)ρ2sinϕdϕdρdθType ρϕ:R∭f(ρ,ϕ,θ)dV=∫ρ1ρ2∫g(ρ)h(ρ)∫p(ρ,ϕ)q(ρ,ϕ)f(ρ,ϕ,θ)ρ2sinϕdθdϕdρType ϕρ:R∭f(ρ,ϕ,θ)dV=∫ϕ1ϕ2∫g(ϕ)h(ϕ)∫p(ρ,ϕ)q(ρ,ϕ)f(ρ,ϕ,θ)ρ2sinϕdθdρdϕType θϕ:R∭f(ρ,ϕ,θ)dV=∫θ1θ2∫g(θ)h(θ)∫p(θ,ϕ)q(θ,ϕ)f(ρ,ϕ,θ)ρ2sinϕdρdϕdθType ϕθ:R∭f(ρ,ϕ,θ)dV=∫ϕ1ϕ2∫g(ϕ)h(ϕ)∫p(θ,ϕ)q(θ,ϕ)f(ρ,ϕ,θ)ρ2sinϕdρdθdϕ
Notice that each integral can have limits which depend only on the variables
outside that integral.
Find the volume of the orange wedge given in spherical
coordinates by 0≤ρ≤5, 0≤θ≤6π, and
0≤ϕ≤π.
This is how the wedge looks:
This is a straightforward application of integration in spherical
coordinates. Our integral with limits is as follows:
V=∫0π/6∫0π∫05ρ2sinϕdρdϕdθ=∫0π/6∫0π[3ρ3]ρ=05sinϕdϕdθ=3125∫0π/6∫0πsinϕdϕdθ=3125∫0π/6∫0πsinϕdϕdθ=3125∫0π/6[−cosϕ]ϕ=0πdθ=3125(2)∫0π/6dθ=3250⋅6π=9125π
We should have known this answer. Since 6π is 121
of a full circle 2π, the wedge is 121 of a sphere and its
volume is 121 of the volume of a sphere of radius 5:
V=12134π(5)3=9125π.
However, it is useful to write down the triple integral for the volume so we
can use it to find the integrals for average value, centroid, mass, and
center of mass.
Find the mass of the wedge in the previous example given that
its density is δ=ρ
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