Basically the answer should be a perfect zero (according to theory and also verified through hand calculation), so why this software is giving such high error.
Please tell me if I am doing something wrong as soon as possible.
Because you are numerically evaluating the integral
The precision with which Mathcad calculates numerical evaluation is in the order of 10^-17.
You're doing nothing wrong. You could set the 'zero tolerance' of numerical calculation to a higher value (e.g. 10^-10) this would show all numbers x with |x|<10^-10 as 0. But it would obscure results that are actually meant to be e.g. 10^-12.
This is not a 'problem'. Just be aware that all calculations go with a certain uncertainty/accuracy.
Simply calculate 1/2-1/3-1/6 and see what comes out.Mathcad will tell you:
MsExcell will tell you 0, until you multiply the result with 10^18, then it spits out 27.76
If you want exact answers, use a symbolic engine (Mathcad, Maple, Mathematica, Maxima).
P.S. If you want to know why the 'error' is in the order of 10^-17, have a look at:
It's numerical evaluation precision error. (10^-17 is a large error?)
Symbolic evaluation gives you zero: