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Question Number 183216 by mr W last updated on 23/Dec/22

Commented by mr W last updated on 23/Dec/22

Commented by mr W last updated on 23/Dec/22

the general case for sphere (red) cut  by a cylinder (green)

thegeneralcaseforsphere(red)cutbyacylinder(green)

Answered by mr W last updated on 23/Dec/22

given: R, r, a  find the volume of the portion of  the sphere cut by the cylinder  b^2 =a^2 +r^2 −2ar cos θ  z=(√(R^2 −b^2 ))=(√(R^2 −a^2 −r^2 +2ar cos θ))  a sin θ=(a−r cos θ) tan ϕ  ⇒ϕ=tan^(−1) (((a sin θ)/(a−r cos θ)))  2bdb=2ar sin θ dθ  dS=2ϕbdb=2ar tan^(−1) (((a sin θ)/(a−r cos θ))) sin θ dθ  V=2∫_0 ^π zdS  V=4arR∫_0 ^π (√(1−((a/R))^2 −((r/R))^2 +((2ar)/R^2 ) cos θ)) tan^(−1) (((sin θ)/(1−(r/a) cos θ))) sin θ dθ  let λ=(a/R), μ=(r/R)  ⇒V=4μλR^3 ∫_0 ^π (√(1−μ^2 −λ^2 +2μλ cos θ)) tan^(−1) (((sin θ)/(1−(μ/λ) cos θ))) sin θ dθ

given:R,r,afindthevolumeoftheportionofthespherecutbythecylinderb2=a2+r22arcosθz=R2b2=R2a2r2+2arcosθasinθ=(arcosθ)tanφφ=tan1(asinθarcosθ)2bdb=2arsinθdθdS=2φbdb=2artan1(asinθarcosθ)sinθdθV=20πzdSV=4arR0π1(aR)2(rR)2+2arR2cosθtan1(sinθ1racosθ)sinθdθletλ=aR,μ=rRV=4μλR30π1μ2λ2+2μλcosθtan1(sinθ1μλcosθ)sinθdθ

Commented by manxsol last updated on 23/Dec/22

Thanks, Sir W,applause   for your effort, very clear

Thanks,SirW,applauseforyoureffort,veryclear

Commented by mr W last updated on 23/Dec/22

thank you too!

thankyoutoo!

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