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 is h of x equals cos to the 4th x times sin of x on the range
0 to pi.
The formula that we're going to be using for the average value is 1 over b minus a which
gives us the average part on the range of the function and we're going to be integrating
on the limits of the function a to b.
In this case they've given us our range a to b.
It's 0 to pi.
So 0 is a and pi is b.
And the first thing that we’re going to do is plug in our limits of integration 0
and pi.
So again we're plugging in pi for b as you can see here and 0 for a and then we also
plug them in as our limits of our integration here and we take our function h of x and put
it into the integral here.
So that sets us up and now it's a matter of simplifying so that we can get to a point
where we can integrate.
So obviously pi minus 0 out in front here just becomes pi.
The best way to write cos to the 4 of x is as it's written here in the integral.
However, it’s easier to work with it for me when I move it outside.
I can say cos of x and it's all raised to the 4th power which is still technically correct
but the best way to write it is with the 4th after the cos.
The reason I’m going to do this is because this problem is going to require us to use
u-substitution to solve the integral and we’re going to substitute u for cos of x so it's
easier to have this 4th out here for me.
So we're going to set u equals cos of x and the reason that we pick cos of x instead of
sin of x is because remember when we're doing u-substitution, we find u and then we take
the derivative du and the derivative of cos of x is negative sin of x which might allow
us to cancel out this sin of x here.
So that's what I'm hoping for and that's why I picked cos of x to be u.
So we set u equal to cos x and then we take the derivative of u, du and the derivative
of cos x is negative sin of x.
So we have negative sin of x dx here.
We always write the dx when we take the derivative du.
And then we're going to solve for dx by dividing both sides by negative sin of x.
When we do that, we get dx equal to du divided by negative sin of x.
And now that we've done that, what we're going to do is put in du over negative sin x for
dx and u for cos of x.
And hopefully, that will allow us to simplify the integral.
So you can see that if we put u in here for cos of x and here's our 4th power and we put
du over negative sin of x in for dx.
And as you can see, we're already going to get this two sin of x terms here to cancel.
We’re just going to have to pull out this negative 1 in the denominator out in front.
So this negative 1 is going to come out in front of the integral here so we'll end up
with negative 1 over pi and then we’re just left with u to the 4th du in the integral
which is really easy to integrate.
So when we integrate, we'll have this negative 1 over pi which will stay out in front and
as always with integration we add 1 to the exponent so 4 plus 1 gives us 5 and then we
divide the coefficient which is an implied 1 here by the new exponent 5 so we get a 1/5
coefficient out in front of the u and then we'll be evaluating from 0 to pi.
So before we can evaluate, we have to plug back in for u.
Remember we said u was equal to cos of x, so you plug cos of x back in for u and while
we're at it, we move the 5th power back in front of the x here in front of the cos because
that's technically the right way to write it.
So we plugged back in for u and now we evaluate from 0 to pi.
The first thing that we do is plug in pi here to 1/5 cos to the 5th of x.
Then we subtract and then we plug in 0.
Whenever we're evaluating definite integrals, we plug in the top number then we always subtract
and then plug in the bottom number.
So we plug that in and now we're going to start simplifying.
cos to the 5th of pi.
Basically you're looking at cos of pi and what you need to do is grab your Unit Circle.
cos of pi will be the angle along the x-axis and at the angle pi.
You're looking for the x-coordinate because when you're evaluating cos of an angle on
the unit circle, we look at the x-coordinate.
sin of the angle on the unit circle, we have to look at the y-coordinate.
So we're looking at the x-coordinate and along the x axis there, the x-coordinate is equal
to negative 1.
So we plug in negative 1 for cos of pi and we keep our 5th power here on the negative
1.
And then cos of 0, again looking at the Unit Circle, we’ll be looking at the place where
the angle is equal to 0 and cos of that angle again is the x-coordinate at that point
which is 1.
So cos of 0 is 1, we keep our 5th power.
And now negative 1 to the 5th is negative 1 so negative 1 times 1/5 gives us a negative
1/5 here for this first term.
And 1 to the 5th power is simply 1.
So we have a negative 1/5 times 1 stays a negative 1/5.
So when we combine everything inside the parenthesis, negative 1/5 minus another 1/5 gives us a
negative 2/5 here.
And now it's simple enough that we can combine these two because we have a negative and a
negative we get a positive 2/5 pi and that is our final answer.
That’s the average of this function h of x over the range 0 to pi.
I hope that helped you guys and I’ll see you in the next problem.
Bye.