Question

Why isn't the first one true?

Answer 1

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).

Answer 2

Answer:

  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.