Wallis' Formula (Part 1) by J.V.Viola

Summary

This transcript discusses the application of the wireless formula to solve integrals involving trigonometric functions like sine and cosine. It outlines general forms and examples of integrals, focusing on integrals from 0 to π/2 and their various transformations. The script provides detailed steps for calculating specific examples, including integration of sine raised to even powers, sine-cosine products, and higher-degree terms. The examples aim to illustrate how the wireless formula can simplify complex integrals, making the process more manageable through systematic steps and algebraic manipulations.

Takeaways

• 😀 The wireless formula is used to solve integrals involving trigonometric functions raised to specific powers, such as sine and cosine.
• 😀 The general form of the wireless formula includes an integral of sine^m(x) * cosine^n(x) from 0 to pi/2.
• 😀 The formula involves multiple factors, including m and n terms, which are used in calculating the integral, and their relationship to the Gamma function.
• 😀 The Gamma function is important for handling the differences and adjustments in the formula, especially when m and n are even or odd.
• 😀 For integrals of sine^m(x), the formula simplifies to a specific equation, and this equation involves terms such as (m-1), (m-3), etc.
• 😀 Similarly, for integrals of cosine^n(x), the formula is similar, involving terms such as (n-1), (n-3), etc., and adjusting for even or odd exponents.
• 😀 In practical examples, the wireless formula can be applied to specific values of m and n to calculate integrals with powers of sine and cosine.
• 😀 Example 1 shows how to calculate an integral with sine^6(x) * cosine^4(x), breaking down the formula step by step with substitutions.
• 😀 Example 2 demonstrates the use of the formula for sine^2(y) * cosine^4(y), showing how to apply the formula with specific values for m and n.
• 😀 Example 3 involves a substitution to change the limits of integration from x to u, using the substitution u = 2x and adjusting the limits accordingly.
• 😀 Example 4 demonstrates how to handle a sine^3(x) integral by using a substitution method, simplifying the integral and applying the wireless formula.

Q & A

What is Wally's formula and how is it used?

Wally's formula is used to evaluate integrals involving powers of sine and cosine functions. It provides a way to compute integrals of the form ∫[0, π/2] sin^m(x) * cos^n(x) dx by using specific relationships between the values of m and n, expressed in terms of Gamma functions and certain constants.

How does the general form of Wally's formula work?

The general form of Wally's formula involves using values for m and n in the integral. The integral from 0 to π/2 of sin^m(x) * cos^n(x) dx is expressed through the equation: (m-1)(m-3)(m-5)... for m and (n-1)(n-3)(n-5)... for n, all divided by (m+n)(m+n-2)(m+n-4)... with specific conditions for alpha depending on whether m and n are even or odd.

What does the constant 'alpha' represent in Wally's formula?

Alpha is a constant that takes the value of π/2 if both m and n are even. If m or n is odd, alpha equals 1. This constant helps adjust the formula based on the parity of the exponents of sine and cosine.

What is the significance of the 'Gamma function' in Wally's formula?

The Gamma function plays a crucial role in Wally's formula for simplifying the products of terms like (m-1)(m-3)... and (n-1)(n-3)... which appear in the formula. It provides a way to handle the factorial-like expressions and simplifies the computation.

What steps are involved in using Wally's formula for the integral ∫[0, π/2] sin^6(x) * cos^4(x) dx?

To use Wally's formula for this integral, first calculate the terms (m-1)(m-3)(m-5) for m=6 and (n-1)(n-3)(n-5) for n=4. Then divide by the appropriate terms involving m+n and apply the value of alpha (which is π/2 for both even m and n). The result is simplified to give the final integral value.

What is the outcome of applying Wally's formula to the example ∫[0, π/2] sin^6(x) * cos^4(x) dx?

The result of applying Wally's formula to this integral is 3π/512, after calculating the necessary constants and simplifying the terms.

How is the integral ∫[0, π/2] sin^2(y) * cos^5(y) dy solved using Wally's formula?

For this integral, the terms for m=2 and n=5 are calculated: (2-1)(2-3) and (5-1)(5-3)(5-5). After applying Wally's formula, the result is simplified to 8/105.

What role does the substitution u=2x play in solving ∫[0, π/4] cos^6(2x) dx?

The substitution u=2x is used to simplify the integral. By changing the limits accordingly (from 0 to π/2 for u), the integral becomes easier to handle, and the formula is then applied to compute the result.

What is the result of solving ∫[0, π/4] cos^6(2x) dx using Wally's formula?

After applying Wally's formula and simplifying the terms, the result of this integral is 5π/64.

How is the integral ∫[0, 3π/2] sin^3(x/3) dx evaluated using Wally's formula?

This integral is solved by substituting u=x/3, adjusting the limits accordingly, and then applying Wally's formula for the integral ∫[0, π/2] sin^3(u) du. The result is simplified to 3 times the integral value, yielding the final answer.