Problem 7 - Calculus: Early Transcendentals

Question

Find the average value of the function on the given interval. \( h(x) = \cos^4 x \sin x \) , \( [0, \pi] \)

Step-by-Step Solution

Answer

The average value is \( \frac{2}{5\pi} \).

Step 1: Understand the problem

To find the average value of a function on a given interval, we need to integrate the function over the interval and then divide by the length of the interval. In this problem, the function is \( h(x) = \cos^4 x \sin x \) and the interval is \([0, \pi]\).

Step 2: Determine the formula for average value

The average value \( \overline{f} \) of a function \( f(x) \) on the interval \([a, b]\) is given by: \[ \overline{f} = \frac{1}{b-a} \int_{a}^{b} f(x) \, dx \]. In this case, the formula becomes \[ \overline{h} = \frac{1}{\pi - 0} \int_{0}^{\pi} \cos^4 x \sin x \, dx \].

Step 3: Set up the integral

The integral we need to solve is \( \int_{0}^{\pi} \cos^4 x \sin x \, dx \). We will use substitution to simplify the integration process.

Step 4: Choose a substitution

Let \( u = \cos x \), then \( du = -\sin x \, dx \) or \( -du = \sin x \, dx \). When \( x = 0, u = 1 \) and when \( x = \pi, u = -1 \). The integral becomes \( \int_{1}^{-1} u^4 (-du) \) which simplifies to \( \int_{-1}^{1} u^4 \, du \).

Step 5: Integrate the function

Now we integrate \( u^4 \) from \(-1\) to \(1\): \[ \int_{-1}^{1} u^4 \, du = \left[ \frac{u^5}{5} \right]_{-1}^{1} = \frac{1^5}{5} - \frac{(-1)^5}{5} = \frac{1}{5} + \frac{1}{5} = \frac{2}{5} \].

Step 6: Calculate the average value

Now that we have the integral result, we calculate the average value: \[ \overline{h} = \frac{1}{\pi} \times \frac{2}{5} = \frac{2}{5\pi} \].

Key Concepts

Integration

Integration is a key concept in calculus that involves calculating the area under the curve of a function. In this exercise, we use integration to find the average value of a trigonometric function over an interval. The basic idea is to accumulate or "add up" tiny slices of area, under the curve of the function, across a specified range.
To find the average value of a function like \( h(x) = \cos^4 x \sin x \) over an interval \([0, \pi]\), we need to compute the integral of that function. This allows us to determine the total area under the curve from \( x = 0 \) to \( x = \pi \), which is then divided by the length of the interval.

• The integral provides the net "accumulated" result over the interval.
• We represent the integral mathematically using the integral sign \( \int \), the function, and the limits of integration from \( a \) to \( b \).

By calculating the integral \( \int_{0}^{\pi} \cos^4 x \sin x \, dx \), we get an essential component for determining the function's average over the specified range.

Substitution Method

The substitution method is a technique used in integration to simplify complex integrals. It's similar to the chain rule in differentiation and involves choosing a substitution that transforms the integrand into a simpler form.
In our given exercise, the function \( h(x) = \cos^4 x \sin x \) is complex, and we choose a substitution \( u = \cos x \) to make the integration process more manageable.

• Choosing an effective substitution helps rewrite the integral in terms of the new variable \( u \), making it easier to integrate.
• We differentiate our chosen substitution which gives us \( du = -\sin x \, dx \), allowing us to express \( \sin x \, dx \) in terms of \( du \).
• The limits of integration also change accordingly when \( u = \cos x \), leading from \( u = 1 \) to \( u = -1 \) as \( x \) goes from 0 to \( \pi \).

By converting \( \int_{0}^{\pi} \cos^4 x \sin x \, dx \) into \( \int_{-1}^{1} u^4 \, du \), integration becomes much more straightforward.

Trigonometric Functions

Trigonometric functions like sine, cosine, and tangent are fundamental in mathematics, describing relationships in right-angled triangles and oscillatory motions such as waves.
In our exercise, we're dealing with \( \cos^4 x \sin x \). These functions often appear in integrals, and their cyclical properties make integration a frequent topic.

• Trigonometric functions are periodic, meaning they repeat values in a regular interval, which is crucial when defining limits for integration.
• We use relationships between trigonometric identities, like \( \sin \) and \( \cos \), to simplify complex expressions, especially in integration problems.
• Understanding trigonometric functions' behavior under transformation (e.g., from \( x \) to \( u \) in substitution) is essential for successfully solving such integrals.

By leveraging trigonometric identities and substitutions, we can handle complex expressions like \( \cos^4 x \sin x \) more effectively, leading to efficient integration solutions.