Let f (x) and g (x) be the functions graphed below.
Find .
Answer
The area between f (x) and the x-axis is a triangle.
so
Example 2
Find
Answer
The triangle moved a bit to the right, but still has the same size and shape.
Example 3
Find
Answer
First we need to remember that f (x) ⋅ g (x) means for each value of x, we take f (x) and g (x) and multiply them together.
For example,
f (0) ⋅ g (0) = 2 ⋅ 0 = 0.
Let x be any value less than 1. Then g (x) = 0, so f (x) ⋅ g (x) = 0 also.
Now let x be any value between 1 and 2. Then f (x) = 0, so f (x) ⋅ g (x) = 0.
The function f (x) ⋅ g (x) is the zero function.
The integral of the zero function is zero, so
Example 4
Does ?
Answer
No. We saw that
but
The integral of the product is not the product of the integrals.
The properties having to do with inequalities will get used more after we get to improper integrals. For now, we'll focus on the other properties we've seen so far.
Example 5
Assume f and g are integrable functions.
If and then
Answer
Since the integral of a sum is the sum of the integrals,
We can pull out the constants:
Finally, substitute in the values of the integrals and to get
Example 6
If, f ( x ) is an even function, and g ( x ) is an odd function, then
Assume f and g are integrable functions.
Answer
We start the same way as the last problem, by splitting up the integral and pulling out the constants:
Since f is even, we know that
Since g is odd, we know that
Putting these values in where we left off gives us
Example 7
If f is odd then
Answer
Using the property
with a = -3, b = 1, and c = 2, we can combine the first two terms of this expression:
Using the same property again, this time with a = -3, b = 2, and c = 3, we can combine the remaining two terms:
Since we're assuming f is odd, this integral is equal to 0.
Example 8
If then
Answer
Again, we can take the necessary steps in a couple of different orders.
.
?
and 
then
and 
to get
, f ( x ) is an even function, and g ( x ) is an odd function, then
then
to get
