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ʃ[0,2π]dθʃ[0,1]ρ²√(1+4ρ²)·ρdρ
=πʃ[0,1]ρ²√(1+4ρ²)d(ρ²)
=πʃ[0,1]t√(1+4t)dt(t=ρ²)
=(π/16)ʃ[0,1]4t√(1+4t)d(4t)
=(π/16)ʃ[0,1][(1+4t)√(1+4t)-√(1+4t)]
d(4t)
=(π/16)ʃ[0,1][(1+4t)³⸍²-(1+4t)¹⸍²]d(1+4t)
=(π/16)[2(1+4t)⁵⸍²/5-2(1+4t)³⸍²/3]|[0,1]
=(π/16)[2(5⁵⸍²-1)/5-2(5³⸍²-1)/3]
=(π/120)[3(5⁵⸍²-1)-5(5³⸍²-1)]
=(π/120)[3(25√5-1)-5(5√5-1)]
=(π/120)(50√5+2)
=π(25√5+1)/60.
=πʃ[0,1]ρ²√(1+4ρ²)d(ρ²)
=πʃ[0,1]t√(1+4t)dt(t=ρ²)
=(π/16)ʃ[0,1]4t√(1+4t)d(4t)
=(π/16)ʃ[0,1][(1+4t)√(1+4t)-√(1+4t)]
d(4t)
=(π/16)ʃ[0,1][(1+4t)³⸍²-(1+4t)¹⸍²]d(1+4t)
=(π/16)[2(1+4t)⁵⸍²/5-2(1+4t)³⸍²/3]|[0,1]
=(π/16)[2(5⁵⸍²-1)/5-2(5³⸍²-1)/3]
=(π/120)[3(5⁵⸍²-1)-5(5³⸍²-1)]
=(π/120)[3(25√5-1)-5(5√5-1)]
=(π/120)(50√5+2)
=π(25√5+1)/60.
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