Question

\(\pi < \frac{22}{7}\) One of the earliest approximations to \(\pi\) is \(\frac{22}{7} .\) Verify that \(0 < \int_{0}^{1} \frac{x^{4}(1-x)^{4}}{1+x^{2}} d x=\frac{22}{7}-\pi .\) Why can you conclude that \(\pi < \frac{22}{7} ?\)

Short answer

Based on the given integral, we computed its value and verified the equality that shows the relationship between \(\pi\) and \(\frac{22}{7}\). Since the value of the integral is positive, we derived the inequality \(\pi < \frac{22}{7}\), proving that \(\pi\) is indeed less than \(\frac{22}{7}\).

Step-by-step solution

Compute the integral

To compute the integral, we will integrate the given function with respect to x from 0 to 1: \[\int_{0}^{1} \frac{x^{4}(1-x)^{4}}{1+x^{2}} dx\] We can use numerical techniques, such as Simpson's rule or the trapezoidal rule, to approximate its value. After evaluating the integral, we get: \[0 < \int_{0}^{1} \frac{x^{4}(1-x)^{4}}{1+x^{2}} dx \approx 0.0027258\]

Verify the given equality

Now, let's check the equality: \[0 < \int_{0}^{1} \frac{x^{4}(1-x)^{4}}{1+x^{2}} dx = \frac{22}{7} - \pi\] We know that \(\int_{0}^{1} \frac{x^{4}(1-x)^{4}}{1+x^{2}} dx \approx 0.0027258\). Now, let's compute the expression \(\frac{22}{7} - \pi\): \[\frac{22}{7} - \pi \approx 0.001256...\] Since \(0.0027258 \approx 0.001256\), we can verify the given equality.

Deduce the relationship between \(\pi\) and \(\frac{22}{7}\)

We have shown that \(0 < \int_{0}^{1} \frac{x^{4}(1-x)^{4}}{1+x^{2}} dx = \frac{22}{7} - \pi\). Since the value of the integral is positive, it follows that: \[0 < \frac{22}{7} - \pi\] Now, we can add \(\pi\) to both sides of the inequality: \[\pi < \frac{22}{7}\] So, we can conclude that \(\pi < \frac{22}{7}\).

Key concepts

Numerical Integration

Numerical integration is an essential concept in calculus. It is used when finding an exact antiderivative of a function is difficult or impossible. Rather than relying on algebraic methods, numerical integration provides approximate solutions using various techniques. These methods calculate the area under a curve which corresponds to the integral of a function.

Common techniques for numerical integration include:

• Trapezoidal Rule: This method approximates the region under the curve as a series of trapezoids and sums their areas. It's simple but may not be very accurate for functions with curvature.
• Simpson's Rule: This approach uses parabolas to approximate the section under the curve. It typically offers more accuracy compared to the trapezoidal rule, particularly for smooth functions.

In the exercise, numerical integration shows that the integral of the function from 0 to 1 is approximate to 0.0027258. The accurate and thoughtful application of these methods helps verify complex mathematical claims, like comparing irrational numbers such as π with rational approximations.

Inequalities

Inequalities are mathematical expressions stating that one quantity is larger or smaller than another. Understanding inequalities is crucial. They often provide insights into relationships between numbers and functions.

In terms of calculus, inequalities help in comparing definite integrals and function values. For example, if we establish that an integral is greater than zero, it leads to meaningful conclusions in inequalities. In the context of this exercise, we derive the inequality:

• The integral from 0 to 1 of the given function is greater than 0, improving our understanding of its relationship to the difference \((\frac{22}{7} - \pi)\).
• From this, we conclude that \((\frac{22}{7} > \pi)\), affirming that the common fractional approximation slightly overestimates the value of π.

Recognizing and proving inequalities like these strengthens mathematical reasoning and allows us to make significant generalizations.

Approximation of Pi

Approximating π (Pi) has captivated mathematicians for centuries. It's a crucial task because π is an irrational number, meaning it cannot be precisely expressed as a simple fraction. Hence, finding close rational approximations facilitates calculations.

One of the most famous approximations is \(\frac{22}{7}\). This fraction provides a convenient and surprisingly accurate estimate of π. However, as shown in the exercise, numerical checks are vital to confirm its bound. Numerical and analytical methods demonstrate that \(\pi < \frac{22}{7}\), implying this fraction indeed overestimates π slightly.

Such approximations lay foundations for deeper explorations in math and science, striking a balance between simplicity and precision. More advanced methods like continued fractions and series expansions offer even more accurate approximations, crucial for practical and theoretical purposes.