Finding the Average Value of the Function h(x) = 6cos⁴(x)sin(x) on the Interval [0, π]

As an excellent high school mathematics teacher, I'm excited to guide you through the process of finding the average value of the function h(x) = 6cos⁴(x)sin(x) on the interval [0, π]. This exercise will not only help you develop a deeper understanding of trigonometric functions but also strengthen your skills in calculus.

Understanding the Function

The function h(x) = 6cos⁴(x)sin(x) is a product of two trigonometric functions: cos⁴(x) and sin(x). The cosine function, raised to the power of 4, represents the amplitude of the oscillation, while the sine function determines the shape of the oscillation.

Calculating the Average Value

To find the average value of the function h(x) on the interval [0, π], we need to integrate the function over the interval and then divide the result by the length of the interval.

The formula for the average value of a function f(x) on the interval [a, b] is:

Average value = (1 / (b - a)) * ∫(a to b) f(x) dx

In our case, the interval is [0, π], so we have:

Average value = (1 / (π - 0)) * ∫(0 to π) 6cos⁴(x)sin(x) dx

Simplifying the expression, we get:

Average value = (1 / π) * ∫(0 to π) 6cos⁴(x)sin(x) dx

Now, let's solve the integral:

∫(0 to π) 6cos⁴(x)sin(x) dx = 6 * ∫(0 to π) cos⁴(x)sin(x) dx

To solve this integral, we can use integration by parts:

Let u = cos⁴(x) and dv = sin(x) dx

Then, du = -4cos³(x)sin(x) dx and v = -cos(x)

Applying the integration by parts formula, we get:

∫(0 to π) 6cos⁴(x)sin(x) dx = 6 * [-cos⁴(x) + 4∫(0 to π) cos²(x) dx]

Evaluating the integral from 0 to π, we have:

∫(0 to π) 6cos⁴(x)sin(x) dx = 6 * [-cos⁴(π) + cos⁴(0) + 4∫(0 to π) cos²(x) dx]

Simplifying the expression, we get:

∫(0 to π) 6cos⁴(x)sin(x) dx = 6 * [1 + 1 + 4 * (π/2)]

Substituting the result into the formula for the average value, we have:

Average value = (1 / π) * 6 * [1 + 1 + 2π] = 6 * (2 + π) / π

Therefore, the average value of the function h(x) = 6cos⁴(x)sin(x) on the interval [0, π] is 6 * (2 + π) / π.

I hope this detailed explanation has helped you understand the process of finding the average value of a function. If you have any further questions or need additional assistance, feel free to ask!