Converting a rieman sum to definite integral

Question

∑ (1)/(sqrt(4n²-i²)) 

Answer 1

lim_n→∞ ∑_i=0^n-1 1/√(4n²-i²) = lim_n→∞ (1/n)∑_i=0^n-1 1/√(4-(i/n)²) = ∫₀¹ 1/√(4-x²) dx.

 

Note that ∫₀¹ 1/√(4-x²) dx = sin^-1(x/2)|₀¹ = sin^-1(1/2) = π/3.