Why isn't the first one true?

Step-by-step explanation:

The integral represents the area under a curve.  You can think of it as a sum.  Just because the sum of f(x) is greater than the sum of g(x), does not mean that each f(x) term is greater than the corresponding g(x) term.

For example, 3 + 2 + 1 = 6, and 1 + 3 + 1 = 5.  The first sum is greater than the second sum, but the middle term in the first sum (2) is less than the middle term of the second sum (3).

  I. requires a function relationship everywhere that may not exist.

Step-by-step explanation:

The integral gives the area under the curve. It says nothing about the function value, except on average.

Consider the case where [a, b] = [0, 10] and ...

  g(x) = -0.1

  f(x) = -2 + 25·δ(x)

(The impulse function δ(x) has unit area and is zero everywhere except x=0.)

In this scenario, the only place where f(x) > g(x) is at x=0.