Hi everyone.
Welcome back to integralcalc.com.
Today we're going to be doing another average value of a function problem.
The function that we're given in this case is g of x equals cos of x and we're asked
to find the average value of that function over the range 0 to pi/2.
So the formula that we’re going to be using to do this is the following, which is a formula
we're always going to use when we're finding the average value of a function over a given
range.
And the first thing that we do when we use this formula, is plug in values for both a
and b.
So we're given a range from a to b.
In other words, we're going to be plugging in 0 for a and pi/2 for b.
And we do that both in the fraction out in front of the integral, and as our limits of
integration in here.
So we plug those two in and that sets up our integral for us.
Then what we do is simplify the fraction and integrate.
So when we simplify the fraction out in front here, pi/2 minus 0 of course is pi/2.
Whenever you have 1 divided by a fraction, when you simplify it, the result will be the
inverse of that fraction.
So instead of 1 divided by pi/2 we end up with 2/pi.
So we have 2/pi and that's out in front as our fraction.
The integral of cos of x is sin x.
And then of course we’re going to be evaluating on the range 0 to pi/2.
So that’s how we simplify and integrate and then when we evaluate on this range, first
we plug in pi over 2 and then we subtract and plug in 0 for x.
If you have trouble with definite integrals, I have that section of videos on my website.
You can check that out.
You always plug in the top number first and then subtract and then plug in the bottom
number so we end up with sin of pi/2 minus sin of 0.
Now, in order to evaluate sin of pi/2 and sin of 0, we need to use our Unit Circle here.
So let's first look at sin of pi over 2.
The way that we use the Unit Circle to evaluate sin of pi/2 and sin of 0 for that matter is
we're looking for the place where the angle is equal to pi/2 so what we do is we go over
her to our unit circle and we look along the unit circle until we find a place where
we find that the angle is equal to pi/2 and we can see that it's right here and that's
at the point (0, 1) where x is equal to 0 and y is equal to 1.
So because we're looking for sin of this angle, sin of pi over 2, we look at the y-coordinate
of this point.
And the y-coordinate here is positive 1.
So sin of pi/2 heading back over here to our equation is equal to 1.
sin of 0, if we head back out here to the unit circle, again we're looking for the place
where the angle is equal to 0 so we look and we see that it's right here in the x axis.
The angle is equal to 0 and because we’re looking for sin of that angle, we’re looking
at the y-coordinate.
Again we're using the unit circle.
If you're looking for cos, you check out the x-coordinate and if you're looking for sin,
you look at the y-coordinate.
So sin of 0 you can see here, the y-coordinate is 0 so sin of 0 is going to simplify to 0.
Heading back over here to the function, you see we've written in 0 here so obviously this
is going to be 2/pi times 1 which means that our final answer ends up being 2/pi.
2/pi is the average value of the cos function over the range 0 to pi/2.
I hope that helped you guys and I'll see you in the next example.
Bye.