cumsum function question for integration

The code is below.

close all;
clear all;
clc;
format long;

syms t;

inter_val = 0.0001;
tn=0:inter_val:20;
t=tn;
y_cos = sin(t);
int_y_cos=cumsum(y_cos*inter_val);

figure(1);
plot(t, y_cos, 'k');
box on; grid on;

figure(2);
plot(t, int_y_cos, 'r');
box on; grid on;

I want to know why figure(2) didn't come out with cos plot but with offset?

But switch y_cos = sin(t) to y_cos = cos(t), it can give a perfect sin plot?

Thanks,

Answer 1

[Torsten]

cumsum(y_cos*inter_val);

is a Riemann sum for the integral of sin(t) - integrated between 0 and x.

But the integral of sin(t) from 0 to x is 1-cos(x), not cos(x).

Answer 2

[Cris LaPierre]

It has to do with the fact that sin(t) starts at 0 and is positive until pi. cumsum is just adding up the Y values, so it oscillates between 0 and 2. With cos(t), it is only positive for pi/2, so cumsum only gets to half the value before it starts decreasing, or 1. It then is negative for pi, decreasing the value by 2, reaching -1. It therefore oscillates between -1 and 1.