Let f (x) and g (x) be the functions graphed below.
The area between f (x) and the x-axis is a triangle.
The triangle moved a bit to the right, but still has the same size and shape.
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.
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
No. We saw that
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.
Assume f and g are integrable functions.
If and then
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
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
If f is odd then
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.
Again, we can take the necessary steps in a couple of different orders.
Pull out the constant:
Switch the limits and switch the sign:
Substitute to get
Since the limits of integration are the same,
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