Plus, the limits of integration have no distinction between "=" and "<"...nor do they have any distinction between "=" and ">".

If you really want to be more comfortable, you can do the integration from 0 to 1 of f(x) = 3, and THEN add it to the integral of f(x) = 6 for integrating from 1+h to 2.

Then take the limit as h approaches zero. What do you get? You get exactly what you would get if you never even thought about what h is.

Integral of f(x)=3 from 0 to 1 = 3
Integral of f(x)=6 from 1+h to 2 = 6*(2 - (1 + h)) = 6*(1 - h)

Add them up:
3 + 6*(1 - h)

Now, take the limit as h approaches zero. What do you get? Why 3+6 of course, which equals 9. And that is exactly what you would get if you never thought about h.

To show you, suppose:
h=0.1...then 3+6*(1-h) = 8.4
h=0.01...then 3+6*(1-h) = 8.94
h=0.001...then 3+6*(1-h) = 8.994
h=0.0001...then 3+6*(1-h) = 8.9994

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