Problem 17 - Essential Calculus

Question

Use Wallis's Formulas to evaluate the integral. $$ \int_{0}^{\pi / 2} \sin ^{6} x d x $$

Step-by-Step Solution

Answer

The value of the integral \( \int_{0}^{\pi / 2} \sin ^{6} x d x \) is \( \frac{5}{32} \cdot \pi \)

Step 1: Identify the nature of the problem

We have to compute \( \int_{0}^{\pi / 2} \sin ^{6} x d x \). This is the integral of an even power of sine over the interval from 0 to \( \pi/2 \), so we can use Wallis's Formula.

Step 2: Apply Wallis's Formula

To apply Wallis's Formula, we use the fact that for even integers \( n \), \( \int_{0}^{\pi / 2} \sin ^{n} x d x = \int_{0}^{\pi / 2} \cos ^{n} x d x = \frac{(n-1)!!}{n!!} \cdot \frac{\pi}{2} \), where \( (n)!! \) represents the double factorial computed as the product of all positive integers less than or equal to \( n \) that have the same parity (odd or even). Here, we compute \( \int_{0}^{\pi / 2} \sin ^{6} x d x = \frac{(6-1)!!}{6!!} \cdot \frac{\pi}{2} \).

Step 3: Compute the double factorials

Next, compute the double factorials. The double factorial of an odd integer \( n \) is the product of all positive odd integers less than or equal to \( n \), and the double factorial of an even integer is the product of all positive even integers less than or equal to \( n \). For the numerator, \( (6-1)!! = 5!! = 5 \cdot 3 \cdot 1 = 15 \). For the denominator, \( 6!! = 6 \cdot 4 \cdot 2= 48 \). The integral is then \( \frac{15}{48} \cdot \frac{\pi}{2} \).

Step 4: Simplify the expression

Simplify the expression \( \frac{15}{48} \cdot \frac{\pi}{2} = \frac{5}{32} \cdot \pi \)

Step 5: Final answer

So the evaluated integral is \( \frac{5}{32} \cdot \pi \)

Key Concepts

Wallis's Formula

Wallis's Formula is a remarkable tool in calculus, particularly useful for evaluating specific types of integrals. Introduced by John Wallis in the 17th century, it allows us to calculate integrals of the form \( \int_{0}^{\pi/2} \sin^n x \, dx \) or \( \int_{0}^{\pi/2} \cos^n x \, dx \) where \( n \) is an even integer. This is quite handy because these integrals can be tricky to solve using basic methods.

The formula itself is expressed as follows for even integers \( n \):

• \( \int_{0}^{\pi/2} \sin^n x \, dx = \int_{0}^{\pi/2} \cos^n x \, dx = \frac{(n-1)!!}{n!!} \cdot \frac{\pi}{2} \)

Here, the double factorial notation \( (n)!! \) is used, which simplifies the calculations involved. Wallis's Formula offers a systematic way to evaluate such integrals, transforming what could be a challenging calculus problem into more manageable arithmetic computations.

Even Power of Sine

When dealing with integrals involving trigonometric functions like sine, an even power of sine occurs often. In integrals such as \( \int_{0}^{\pi/2} \sin^6 x \, dx \), the presence of an even power (in this case, 6) allows us to apply specific mathematical techniques that simplify the problem.

One such technique involves using trigonometric identities and reduction formulas, which can convert the problem into a form suitable for the application of Wallis's Formula. The property of being an even function also aids in this simplification, as sine raised to an even power results in a symmetric integral that ranges from 0 to \( \pi/2 \), which aligns perfectly with Wallis's approach.

This idea of exploiting symmetry and using powerful formulas can drastically reduce the effort needed to solve integrals that may initially seem complicated, making even powers of sine easier to manage.

Double Factorials

Double factorials provide a concise way to deal with sequences of products in mathematics, especially in the realm of combinatorics and calculus. A double factorial of an integer \( n \), denoted as \( n!! \), is the product of integers in a particular sequence, with a common use being selecting integers from a series either all odd or all even.

For even integers, say \( n \, \text{is}\, 6\), the double factorial (6!!) is calculated as follows:

• \( 6!! = 6 \cdot 4 \cdot 2 = 48 \)

For odd integers, the process is similar, for example, \( 5!! \):

• \( 5!! = 5 \cdot 3 \cdot 1 = 15 \)

Understanding how to compute double factorials is crucial when using Wallis's Formula, as they often appear in the final expressions for integrals. Double factorials simplify sequences of multiplies and divide into manageable chunks, making them essential for calculations involving series or integral evaluations.