To find the average value of the function h(x) = 3cos^4(x)sin(x) on the interval [0, 2π], apply the Average Value Theorem. This gives 1/2π times the integral from 0 to 2π of 3cos^4(x)sin(x) dx. This integral's result is the function's average value over one full cycle.
To find the average value of a function h on the specified interval, you would normally use the Average Value Theorem. This theorem states that the average value of a function f on a closed interval [a,b] is given by 1/(b-a) ∫ from a to b [f(x) dx].
However, the function here is periodic, and they are asking for the average value over one complete cycle. For the function h(x) = 3cos^4(x)sin(x), it is important to note that because the sine and cosine only differ in phase, the average over one complete cycle is the same. So, instead of doing a complicated integration, we can simplify the function to its average over one period.
For a trigonometric function like this, the period is 2π. So, the calculation is 1/(2π-0) ∫ from 0 to 2π [3cos^4(x)sin(x) dx], which simplifies to 1/2π ∫ from 0 to 2π [3cos^4(x)sin(x) dx]. The answer to this integral will give the average value of the function h on the interval [0, 2π].