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lim(n->∞) ∑(i:1->n) √(n^2-i^2) /n^2
=lim(n->∞) (1/n)∑(i:1->n) √(1-(i/n)^2)
=∫(0->1) √(1-x^2) dx
=π/4
let
x=sinu
dx=cosu du
x=0, u=0
x=1, u=π/2
∫(0->1) √(1-x^2) dx
=∫(0->π/2) (cosu) ^2 du
=(1/2)∫(0->π/2) (1+cos2u) du
=(1/2) [u+(1/2)sin2u] |(0->π/2)
=π/4
=lim(n->∞) (1/n)∑(i:1->n) √(1-(i/n)^2)
=∫(0->1) √(1-x^2) dx
=π/4
let
x=sinu
dx=cosu du
x=0, u=0
x=1, u=π/2
∫(0->1) √(1-x^2) dx
=∫(0->π/2) (cosu) ^2 du
=(1/2)∫(0->π/2) (1+cos2u) du
=(1/2) [u+(1/2)sin2u] |(0->π/2)
=π/4
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