Problem 42 - Calculus: Early Transcendentals

Question

Given that \( \displaystyle \int^{\pi}_0 \sin ^4x \, dx = \frac{3}{8} \pi \), what is \( \displaystyle \int^0_{\pi} \sin ^4\theta \, d\theta \)?

Step-by-Step Solution

Answer

The value of \( \int^0_{\pi} \sin^4 \theta \, d\theta \) is \( -\frac{3}{8} \pi \).

Step 1: Understanding the Problem

We are given the integral \( \int^{\pi}_0 \sin ^4x \, dx = \frac{3}{8} \pi \). We need to find the value of the integral \( \int^0_{\pi} \sin ^4\theta \, d\theta \). Notice the order of integration limits is reversed.

Step 2: Reversing the Integration Limits

In general, reversing the limits of an integral changes the sign of the integral. Therefore, we use the property: \( \int^b_a f(x) \, dx = -\int^a_b f(x) \, dx \). Applying this property to our integral: \( \int^0_{\pi} \sin^4 \theta \, d\theta = -\int^{\pi}_0 \sin^4 x \, dx \).

Step 3: Substituting the Given Integral Value

We can substitute the known value of the integral: \( -\int^{\pi}_0 \sin^4 x \, dx = -\frac{3}{8} \pi \).

Step 4: Conclusion of the Calculation

The integral \( \int^0_{\pi} \sin^4 \theta \, d\theta \) is equal to \( -\frac{3}{8} \pi \).

Key Concepts

Integration Limits

When you're dealing with definite integrals, the integration limits are crucial. They define the range over which the function is integrated.
It’s like setting the start and end points for a journey along the curve of the function.Let's consider how they work:

• If you have limits from \(a\) to \(b\), you're integrating from \(a\) to \(b\).
• Reversing these limits, from \(b\) to \(a\), flips the direction. This affects the integral's sign.

For example, if \( \int^b_a f(x) \, dx = c \), then \( \int^a_b f(x) \, dx = -c \).
This rule is handy when you encounter reversed limits in a problem. Just remember that flipping the limits flips the sign of the integral.

Integral Properties

Understanding the properties of integrals can make solving problems much easier. Integrals come with a handy set of rules that can simplify calculations.
Here are a few key properties:

• Linearity: You can split integrals across the addition or subtraction of functions. For example, \( \int (f(x) + g(x)) \, dx = \int f(x) \, dx + \int g(x) \, dx \).
• Scalar Multiplication: A constant factor can be moved outside the integral. For example, \( \int a\cdot f(x) \, dx = a\int f(x) \, dx \).
• Reversed Limits: Flipping the integration limits changes the sign of the integral, as seen in our problem.

Understand these properties and use them to simplify and solve integrals efficiently. They can significantly reduce the complexity of a problem.

Trigonometric Integrals

Trigonometric integrals involve trigonometric functions such as sine or cosine. These types of integrals are common in calculus, and solving them often requires specific strategies.
Here are some key points to note:

• Trigonometric functions are periodic, meaning they repeat values in a regular pattern. This can influence the limits you choose.
• Use identities like \( \sin^2(x) + \cos^2(x) = 1 \) to simplify the integrals when possible.
• Common techniques for solving trigonometric integrals include substitution and using specific integral formulas.

In the given problem, we deal with \( \sin^4(x) \), a higher power of sine. Often, such problems can be tackled by reducing the power using trigonometric identities.
Mastering these techniques and understanding the behavior of trigonometric functions can make solving such integrals straightforward.