A simple question: Input problem

Question

I have a simple problem. First I define the value of variables, then I calculate the expression as the following:

Code 1:

a = 10^-6; 
b = 10^-3;
c = 1;
d = 0.1;
s = -d Sqrt[b^2 + c^2];

Integrate[(w E^(-w/a))/(w - s), {w, 0, ∞}]
-3.744368975478385*10^43415

As you see the result is so large. But if I manually insert the value of s (-0.1) into the expression, I would get zero, which is is close to the correct answer:

Code 2:

a = 10^-6; 
b = 10^-3;
c = 1;
d = 0.1;
s = -d Sqrt[b^2 + c^2];

Integrate[(w E^(-w/a))/(w + 0.1), {w, 0, ∞}]
0.

What is the problem with the first code?

Answer 1

In your inputs

a = 10^-6; 
b = 10^-3;
c = 1;
d = 0.1;
s = -d Sqrt[b^2 + c^2]

-0.1

this result is approximated for display. You can see the complete result by placing your cursor in front of the -0.1 and pressing the space bar.

Alternatively

InputForm[s]

-0.1000000499999875

Edit

With s = -d Sqrt[b^2 + c^2] the integral calculation yields ...

in Mathematica 7.0.1.0 : 9.99979500615538*10^-12

in Mathematica 10.3.1 : -3.744368975619640*10^43415

Answer 2

To sum up what we have been saying in comments, it may sometimes be dangerous to use symbolic solvers (e.g. Integrate) with inexact input (e.g. $d=0.1$). It is better in your case to evaluate the integral symbolically, and then calculate the approximate numerical value:

a = 10^-6;
b = 10^-3;
c = 1;
s = -d Sqrt[b^2 + c^2];
d = 1/10;

Integrate[(w E^(-w/a))/(w - s), {w, 0, Infinity}]

N[%]
(* Out: 9.99979*10^-12 *)

The result is of course very close to zero, but not exactly zero. The problem doesn't lie in how close $s$ is to $-0.1$, but in whether you use exact or approximate input to Integrate, which the function does not handle gracefully in this case.