Problem 31 - Essential Calculus

Question

Approximation of \(P i\) In Exercises 30 and 31 , use Simpson's Rule with \(n=6\) to approximate \(\pi\) using the given equation. (In Section 4.8, you will be able to evaluate the integral using inverse trigonometric functions.) $$ \pi=\int_{0}^{1} \frac{4}{1+x^{2}} d x $$

Step-by-Step Solution

Answer

In order to approximate \(\pi\) using the given integral and Simpson's Rule, it is necessary to divide the interval from 0 to 1 into 6 equal parts and evaluate \(\frac{4}{1+x^{2}}\) for each subinterval's \(x_i\). Substituting these results into Simpson's Rule formula, the numerical approximate value of \(\pi\) will be obtained.

Step 1: Identify the values of 'h', 'a' and 'b'

Firstly, it is noticed that the integral runs from 0 to 1. These are going to be our 'a' and 'b' values. \(a=0\) and \(b=1\). Since we are given that \(n=6\), the value of \(h\) can be calculated by \((b-a)/n\), which is \(h = (1-0)/6 = 1/6\). The total width of our interval is divided evenly into 6 slices.

Step 2: Compute \(y_{i}\)

The \(y_{i}\) values for this problem are computed by evaluating the function \(f(x)=\frac{4}{1+x^{2}}\) at each \(x_{i}\), where \(x_i = a + i*h\) for i=0, 1,...n. We need to calculate 7 function values at \(x_{0}=0\), \(x_{1}=1/6\), \(x_{2}=1/3\), \(x_{3}=1/2\), \(x_{4}=2/3\), \(x_{5}=5/6\) and \(x_{6}=1\).

Step 3: Apply Simpson's Rule and Calculate the Value of \(\pi\)

Next, we substitute these \(\{x_{i}\}\) values, that we've just calculated above, into Simpson's Rule formula. Just remember that even-indexed \(y_{i}\) terms are multiplied by 2, and odd-indexed \(y_{i}\) terms are multiplied by 4. The end terms \(y_{0}\) and \(y_{6}\) are not multiplied by any number. The result of this calculation approximates the value of \(\pi\) in this exercise.

Key Concepts

Numerical Integration

Numerical integration is a powerful tool to estimate the value of definite integrals, especially when finding an exact solution is difficult or impossible. One of the popular methods used is Simpson’s Rule. This method provides an excellent balance between simplicity and accuracy.
Simpson's Rule works by approximating the area under a curve using parabolas rather than straight lines like in other methods such as the Trapezoidal Rule.

• For any continuous function, divide the interval \([a, b]\) into an even number of equally spaced sub-intervals.
• Calculate the width of each sub-interval as \(h = \frac{b-a}{n}\), where \(n\) is the number of sub-intervals.
• Use the formula for Simpson’s Rule to approximate the integral: \[\frac{h}{3}(f(x_0) + 4f(x_1) + 2f(x_2) + \dots + 2f(x_{n-2}) + 4f(x_{n-1}) + f(x_n))\]

By applying this, we can compute complex integrals by breaking them into simpler, manageable parts.

Approximation of Pi

Approximating \(\pi\) through integration is an intriguing application of mathematics. One interesting formula is \(\pi = \int_{0}^{1} \frac{4}{1+x^{2}} \, dx\), which comes from the integral of the function representing a circle.
Here’s why it’s effective:

• The integral \(\int_{0}^{1} \frac{4}{1+x^{2}} \, dx\) covers a quarter of the unit circle in the first quadrant, multiplying by 4 gives the full circle's circumference.
• Using numerical methods like Simpson's Rule allows us to approximate this integral effectively, even if computing by hand.

This integral is exact and elegant, leveraging the properties of a circle and calculus to find one of the most famous constants in mathematics.

Inverse Trigonometric Functions

Inverse trigonometric functions are important in calculus for evaluating integrals related to arcs and angles. They are particularly useful when dealing with problems involving circles and circular functions.
In the context of the given problem, understanding inverse trigonometric functions can improve the approximation process:

• The function \(\frac{4}{1+x^{2}}\) is related to the derivative of the inverse tangent function, \(\arctan(x)\), because \(\frac{d}{dx}[\arctan(x)] = \frac{1}{1+x^2}\).
• This relationship helps in evaluating definite integrals involving circular arcs and provides insights into trigonometric identities and properties.

Mastering these functions not only aids in the precise evaluation of integrals but also deepens the understanding of mathematical relationships and patterns.