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Show that the mass of is 16πkby evaluating the integral:

kdV=0π20π201kρ2sinϕdρdθdϕ

Short Answer

Expert verified

Use spherical coordinates ρ,θ,ϕwhile evaluating dVusing triple integral.

Step by step solution

01

Given Information

Spherical coordinates are ρ,θ,ϕ. We need to prove this by solving given integral.

02

Simplification

Taking LHS.

m=kdV

=0π20π201kρ2sinϕdρdθdϕ

=k0π20π201ρ2sinϕdρdθdϕ

=k0π20π2ρ3310sinϕdθdϕ

=k0π20π213sinϕdθdϕ

role="math" localid="1652246785701" =k30π20π2sinϕdθdϕ

role="math" localid="1652246883750" =k30π2θπ20sinϕdϕ

=k30π2π2-0sinϕdϕ

=kπ6-cosϕπ20

=-kπ6cosπ2-cos0

=-kπ60-1

=πk6

Hence, the mass is16πk

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