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3. Use spherical coordinates to valuate ff, y? dV where E is enclosed by the hemisphere x? + y? + z? < 9 and y 2

3. Use spherical coordinates to valuate ff, y? dV where E is enclosed by the hemisphere x? + y? + z? < 9 and y 2

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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3. Use spherical coordinates to valuate fff, y? dV where E is enclosed by the hemisphere x? + y? + z? < 9 and y >
Transcribed Image Text:3. Use spherical coordinates to valuate fff, y? dV where E is enclosed by the hemisphere x? + y? + z? < 9 and y >
Expert Solution
Step 1

Given that 

∫∫∫Ey2dV and  enclosed by the hemisphere, x2+y2+z2≤9,y≥0

Here to evaluate the integral, convert the cartesian coordinates into Spherical coordinates as

x=ρ cosθ cosϕy=ρ sinθ sinϕz=ρ cosϕ

To find ρ:

x2+y2+z2=ρ2 :dV=ρ2 sinϕ dρ dθ dϕx2+y2+z2=9=32ρ=3

The volume of the region between the spheres is given by,

V=∫cd∫αβ∫abfρ sinϕ cosθ, ρ sinϕ sinθ, ρ cosϕρ2sinϕ dρ dθ dϕ

Where E is the spherical wedge given by

E=ρ,θ,ϕ|a≤ρ≤b,α≤θ≤β,c≤ϕ≤d

The region is defined by

E=ρ,θ,ϕ|0≤ρ≤3,-π2≤θ≤π2,0≤ϕ≤π

 

 

 

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