Problem 62 - Essential Calculus

Question

Use a computer algebra system to evaluate the definite integral. $$ \int_{0}^{\pi / 2} \sin ^{6} x d x $$

Step-by-Step Solution

Answer

The definite integral of \(\sin ^{6} x\) from 0 to \(\pi / 2\) is approximately 0.02083.

Step 1: Apply the reduction formula

The reduction formula, \(\sin ^{2 n} x =\frac{1}{2^n. n!} . \sum _{k=0} ^n \binom {n}{k} (2n-2k)! (-4)^{n-k}\), should be applied first to break down \(\sin ^6 x\) into a simpler form.

Step 2: Use the algebra system to integrate

After breaking down the function using the reduction formula, the integral \(\int _0 ^{\pi /2} \sin ^6 x dx\) can be evaluated. The antiderivative at the limits of integration, from 0 to \(\pi/2\), must then be calculated.

Step 3: Calculate the definite integral

Finally, subtract the antiderivative at the lower limit (0) from the antiderivative at the upper limit \(\pi/2\) . The result will be the value for the definite integral.

Key Concepts

Reduction Formula

Reduction formulas are mathematical tools that help in simplifying the process of integration, especially for trigonometric functions that are raised to a power. In our case, the integral involves the function \( \sin^6 x \). To make the integration process manageable, we use a reduction formula.
The reduction formula for \( \sin^{2n} x \) is:

• \( \sin ^{2 n} x =\frac{1}{2^n n!} \sum_{k=0}^{n} \binom{n}{k} (2n-2k)! (-4)^{n-k} \).

This equation allows us to express powers of sine in a simpler form by expanding it into a sum. It transforms the original function into a series of terms that are easier to integrate. For our specific example, where \( n = 3 \), applying this formula breaks down \( \sin^6 x \) into terms that involve factorials and combinations, making the integration step that follows much simpler.
Always keep in mind that using a reduction formula effectively requires practice and familiarity with the underlying structures of factorials and combinations.

Sin Function

The sine function, \( \sin x \), is one of the basic trigonometric functions and plays a crucial role in evaluating integrals involving angles, like in our definite integral.
As a periodic function, it oscillates between -1 and 1, and it repeats every \( 2\pi \). In calculus, understanding how sine behaves is essential for integration, especially when dealing with powers of sine like \( \sin^6 x \).
When integrating such powers, expressing them through identities or reduction formulas (as discussed already) simplifies the computation.

Properties of the Sine Function

• Periodicity: \( \sin(x + 2\pi) = \sin x \).
• Symmetry: It's an odd function, meaning \( \sin(-x) = -\sin x \).
• Values at key points: \( \sin 0 = 0 \), \( \sin \frac{\pi}{2} = 1 \).

These properties facilitate predictions about the behavior of the integral, especially when dealing with specific bounds like from 0 to \( \pi/2 \), where sine smoothly transitions from 0 to 1, allowing for accurate computation of areas under the curve.

Antiderivative

An antiderivative is, essentially, the reverse process of differentiation. For a given function \( f(x) \), its antiderivative \( F(x) \) satisfies \( \frac{d}{dx}F(x) = f(x) \). This concept is the foundation of integral calculus, as the definite integral derives from evaluating the antiderivative at specific limits.
In evaluating \( \int_{0}^{\pi/2} \sin^6 x \, dx \), after applying the reduction formula, we focus on finding the antiderivative of the resulting expression over the limits.

Steps in Antiderivative Calculation

• Identify the function to integrate, here it could be split into simpler terms using the reduction formula.
• Compute the antiderivative, which involves determining \( F(x) \) such that \( F'(x) = \sin^6 x \).
• Evaluate \( F(x) \) at the upper limit (\( \pi/2 \)) and then the lower limit (0).
• Subtract the two results: \( F(\frac{\pi}{2}) - F(0) \).

Understanding antiderivatives facilitates solving definite integrals, ensuring the area under the curve for trigonometric functions raised to powers can be handled efficiently, even when they appear complex initially.