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Cylindrical Triple Integral Find the Volume?

jk8985

New member
Dec 16, 2013
12
Let E be the solid inside cylinder y^2+z^2=1 and x^2+z^2=1, find the volume of e and the surface area of e
 

MarkFL

Administrator
Staff member
Feb 24, 2012
13,775
Can you show us what you have tried so our helpers know where you are stuck?
 

jk8985

New member
Dec 16, 2013
12
For the second half of the question, I got this. Is it correct?? Also, could you write it out nicely :( I don't really understand what I did.

Can someone let me know if this is correct and if I showed all steps?

Since the the region of integration inside x^2 + y^2 = 1 (and symmetry), convert to polar coordinates:
2 * 4 * ∫(θ = 0 to π/2) ∫(r = 0 to 1) (r dr dθ)/(1 - r^2 cos^2(θ))^(1/2)
= 8 * ∫(θ = 0 to π/2) ∫(r = 0 to 1) 2r (1 - r^2 cos^2(θ))^(-1/2) dr dθ
= 8 * ∫(θ = 0 to π/2) (-1/cos^2(θ)) 2(1 - r^2 cos^2(θ))^(1/2) {for r = 0 to 1} dθ
= 16 ∫(θ = 0 to π/2) (1/cos^2(θ)) [1 - (1 - cos^2(θ))^(1/2)] dθ
= 16 ∫(θ = 0 to π/2) (1 - sin θ) dθ/cos^2(θ)
= 16 ∫(θ = 0 to π/2) (sec^2(θ) - sec θ tan θ) dθ
= 16 (tan θ - sec θ) {for θ = 0 to π/2}
= 16 (sin θ - 1)/cos θ {for θ = 0 to π/2}
= 16 (0 - (-1)), using L'Hopital's Rule as θ→ π/2-
= 16.