Video: Finding the Average Value of a Trigonometric Function on a Given Interval

Video Transcript

Find the average value of the function 𝑓 of 𝑥 is equal to four times the cos of three 𝑥 on the closed interval from negative five 𝜋 by 18 to five 𝜋 by 18.

The question gives us a function 𝑓 of 𝑥 which is a trigonometric function. It wants us to find the average value of this function on the closed interval from negative five 𝜋 by 18 to five 𝜋 by 18. Let’s start by recalling what we mean by the average value of a function on a closed interval. If we have a function 𝑓 of 𝑥 which is continuous on a closed interval from 𝑎 to 𝑏, then we can say the average value of this function over this interval is equal to one divided by 𝑏 minus 𝑎 times the integral from 𝑎 to 𝑏 of 𝑓 of 𝑥 with respect to 𝑥.

In our case, we’re looking for the average value of four cos of three 𝑥 on the interval from negative five 𝜋 by 18 to five 𝜋 by 18. So our function 𝑓 of 𝑥 is four cos of three 𝑥. 𝑎 is negative five 𝜋 by 18. And 𝑏 is five 𝜋 by 18. We now need to check that 𝑓 of 𝑥 is continuous on our closed interval. Our function 𝑓 of 𝑥 is a trigonometric function, and we know all trigonometric functions are continuous across their entire domains. And four times the cos of three 𝑥 is defined for all real values of 𝑥.

So our function 𝑓 of 𝑥 is continuous for all real values. So in particular, it must be continuous on our closed interval. So in this case, we can find the average value of our function 𝑓 of 𝑥 on our closed interval by using our formula. We have 𝑓 average is equal to one divided by five 𝜋 by 18 minus negative five 𝜋 by 18 multiplied by the integral from negative five 𝜋 by 18 to five 𝜋 by 18 of four times the cos of three 𝑥 with respect to 𝑥. And this is a very complicated-looking expression. So let’s simplify this slightly.

First, we see the denominator of our fraction is of the form 𝑎 minus negative 𝑎, which we know is two 𝑎. In this case, 𝑎 is five 𝜋 by 18, so we can simplify the denominator of this fraction to be two times five 𝜋 by 18. And two times five 𝜋 by 18 is five 𝜋 by nine, so we simplified 𝑓 average to the following expression. In fact, we can simplify this further. One divided by five 𝜋 by nine is the same as the reciprocal of five 𝜋 by nine. And, of course, the reciprocal of five 𝜋 by nine is nine divided by five 𝜋. The next thing we’ll do is we’ll take the constant of four outside of our integral.

We can now evaluate our integral by using one of our standard integral rules. For any constant 𝑚, where 𝑚 is not equal to zero, the integral of the cos of 𝑚𝑥 with respect to 𝑥 is equal to the sin of 𝑚𝑥 divided by 𝑚 plus the constant of integration 𝑐. And in this case, our value of 𝑚 is equal to three. So 𝑓 average is equal to 36 divided by five 𝜋 times the sin of three 𝑥 divided by three evaluated at the limits of our integral, negative five 𝜋 by 18 and five 𝜋 by 18. Before we evaluate the limits of this integral, we see we have a shared factor of three in our numerator and our denominator. So we’ll cancel this out.

Now, we evaluate this at the limits of our integral. We get 12 over five 𝜋 multiplied by the sin of three times five 𝜋 by 18 minus the sin of three times negative five 𝜋 by 18. Now, we can just simplify this expression. Three times five 𝜋 by 18 is five 𝜋 by six, and three times negative five 𝜋 by 18 is negative five 𝜋 by six. And now, we can just evaluate these angles. The sin of five 𝜋 by six is one-half, and the sin of negative five 𝜋 by six is negative one-half. This gives us 12 over five 𝜋 times one-half minus negative one-half. And, of course, one-half minus negative one-half is a half plus a half, which is just equal to one. So this gives us our final answer. 𝑓 average is equal to 12 divided by five 𝜋.

Therefore, we’ve shown the average value of the function 𝑓 of 𝑥 is equal to four cos of three 𝑥 on the closed interval from negative five 𝜋 by 18 to five 𝜋 by 18 is equal to 12 divided by five 𝜋.