ƒ(t) = cost,0 ≤ t ≤ x
将所求面积分割为n等份的长方体,每份的底长为(x - 0)/n = x/n
而每份的高为ƒ(x/n),ƒ(2x/n),ƒ(3x/n)...ƒ(kx/n)...ƒ(nx/n)
其中第k个长方体的面积为(x/n)ƒ(kx/n)
k个这样的长方体的总面积为Σ(k=1→n) (x/n)ƒ(kx/n),这是大约的面积
取极限,当底长趋向无限小时,lim(n→∞) Σ(k=1→n) (x/n)ƒ(kx/n) = ∫(0→x) ƒ(t) dt
= lim(n→∞) (x/n) Σ(k=1→n) cos(kx/n)
= lim(n→∞) (x/n)[cos(x/n) + cos(2x/n) + cos(3x/n) + ...+ cos((n - 1)x/n) + cos(nx/n)]
= lim(n→∞) (x/n)(1/2)[cosx - 1 + sinxtan(2n/x)]
= x * (sinx)/x
= sinx