Why the area of the quarter sine wave is more than 1 ?

Question

I want to find the area of the quarter sine wave (0 to pi/2)by summation the little squares
I compared this way with integral

I try it by maple 10.
If n=1000
Sigma (i=0 to n-1) sin [ (i*pi)/(2n) ] *(pi/2n) ) ;the value is less than 1
and it’s in the form of the summation of f(x)*delta x
Integral 0 to (pi/2) sin(x)dx ;the value equal to 1

But if I define n = 10000 and I use this way
Sigma (i=0 to n-1) sin [ (i*pi)/(2n) ] *(pi/2n) ) ;the value is more than 1
Why ?

Answer 1

Since you are doing a left-sum, the sum should be < 1 until n is so large that rounding delivers an answer of 1. Check for errors in your data entry.