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Talking About Two Functions

Now another function is going to join the party. Assume both f and g are integrable functions.

  • The integral of a sum is the sum of the integrals. In symbols,

Also, the integral of a difference is the difference of the integrals:

We've taken whatever weighted area was between g and the x-axis, and stuck that on top of f. If g is negative, in some places we may actually be subtracting area from f. The integral of (f + g) is the integral of f plus the integral of g.

  • If f (x) is smaller than g(x), the integral of f is smaller than the integral of g. In symbols, if

f (x) < g(x) for all x in [a, b]


  • If both f and g are positive, it's clear that there's more area between g and the x-axis than between f and the x-axis:
  • If both f and g are negative, the weighted area between g and the x-axis is closer to 0 than the weighted area between f and the x-axis, so we still have

  • If f is negative and g is positive, then  is negative and  is positive.
  • If f and g are sometimes negative and sometimes positive, we can split the interval [a, b] into sub-intervals on which one of the previous conditions holds, and go from there.
  • Let m and M be constants. If mf(x) ≤ M on [a, b], then  

This is using the last property. We're comparing f to the constant functions m and M. If mf (x)on [a, b], then we know from the previous property that

Since , we have half the inequality explained.

Similarly, if f (x) ≤ M on [a,b] then we know from the previous property that

and we know that .

Sample Problem

If f (x) < 2x, then

We can find  by looking at the graph and getting the area of the triangle:

We conclude that

Sample Problem

If   and , then

Be Careful: The integral of a product is not necessarily the product of the integrals.

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