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

Find the average value of π(π₯) = 4 cos 3π₯ on the interval [β5π/18, 5π/18].

04:03

Video Transcript

Find the average value of the function π of π₯ is equal to four times the cos of three π₯ on the closed interval from negative five π by 18 to five π by 18.

The question gives us a function π of π₯ 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. Letβs start by recalling what we mean by the average value of a function on a closed interval. If we have a function π of π₯ which is continuous on a closed interval from π to π, then we can say the average value of this function over this interval is equal to one divided by π minus π times the integral from π to π of π of π₯ with respect to π₯.

In our case, weβre looking for the average value of four cos of three π₯ on the interval from negative five π by 18 to five π by 18. So our function π of π₯ is four cos of three π₯. π is negative five π by 18. And π is five π by 18. We now need to check that π of π₯ is continuous on our closed interval. Our function π of π₯ is a trigonometric function, and we know all trigonometric functions are continuous across their entire domains. And four times the cos of three π₯ is defined for all real values of π₯.

So our function π of π₯ 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 π of π₯ on our closed interval by using our formula. We have π 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 π₯ with respect to π₯. And this is a very complicated-looking expression. So letβs simplify this slightly.

First, we see the denominator of our fraction is of the form π minus negative π, which we know is two π. In this case, π 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 π 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 weβll do is weβ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 π, where π is not equal to zero, the integral of the cos of ππ₯ with respect to π₯ is equal to the sin of ππ₯ divided by π plus the constant of integration π. And in this case, our value of π is equal to three. So π average is equal to 36 divided by five π times the sin of three π₯ 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 weβ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. π average is equal to 12 divided by five π.

Therefore, weβve shown the average value of the function π of π₯ is equal to four cos of three π₯ on the closed interval from negative five π by 18 to five π by 18 is equal to 12 divided by five π.