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Find the volume of solid of region bounded above by the sphere with equation ρ=2and bounded below by the cone with equationϕ=π3.

Short Answer

Expert verified

The required volume isV=83πunits.

Step by step solution

01

Given Information

The given equations are ρ=2andϕ=π3.

02

Evaluation of limits

We know that

x=ρsinϕcosθ,y=ρsinϕsinθ,z=ρcosϕ

and

ρ=x2+y2+z2,tanθ=yx,cosϕ=zρ,dxdydz=ρ2sinϕdρdϕdθ

Limits of spherical coordinates are

0<θ<2π,0<ϕ<π3,0<ρ<2

To find the volume as per given conditions, we will use spherical coordinates.

03

Calculation of Volume

Required Volume is V=Vdxdydz

V=ϕ=0π/3ρ=02θ=02πρ2sinϕdρdϕdθ

V=ϕ=0π/3sinϕdϕρ=02ρ2dρθ=02πdθ

V=(-cosϕ)0π/3ρ33ρ=0ρ=2θθ=02π

Application of limits yields

V=1-12233{2π}

Hence, V=83πunits

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