Question

Evaluate \(\int_{0}^{1} \frac{d x}{1+x^{2}}\) exactly and show that the result is \(\pi / 4\). Then, find the approximate value of the integral using the trapezoidal rule with \(n=4\) subdivisions. Use the result to approximate the value of \(\pi\).

Short answer

The exact integral value is \(\frac{\pi}{4}\); approximating the integral using trapezoidal rule gives an estimate for \(\pi\).

Step-by-step solution

Identify the Integral

Recognize that the integral to evaluate is \( \int_{0}^{1} \frac{d x}{1+x^{2}} \). This is a standard integral that is related to the inverse tangent function, \( \tan^{-1}(x) \).

Use the Antiderivative of the Given Function

The antiderivative of \( \frac{1}{1+x^2} \) is \( \tan^{-1}(x) \). Therefore, the integral \( \int \frac{d x}{1+x^{2}} = \tan^{-1}(x) + C \), where \( C \) is the constant of integration.

Evaluate the Definite Integral

To find \( \int_{0}^{1} \frac{d x}{1+x^{2}} \), use the Fundamental Theorem of Calculus. Evaluate \( \tan^{-1}(x) \) from 0 to 1: \[ \tan^{-1}(1) - \tan^{-1}(0) = \frac{\pi}{4} - 0 = \frac{\pi}{4}. \] This verifies that the exact value of the integral is \( \frac{\pi}{4} \).

Apply the Trapezoidal Rule with n=4

Divide the interval \([0,1]\) into 4 equal parts so \( \Delta x = \frac{1}{4} \). The x-values are \( x_0 = 0, x_1 = 0.25, x_2 = 0.5, x_3 = 0.75, x_4 = 1 \). Compute the y-values: \( f(x_i) = \frac{1}{1+x_i^2} \) for each \( x_i \).

Calculate y-values at Subdivision Points

Calculate \( y \)-values:- \( f(0) = 1 \)- \( f(0.25) = \frac{1}{1+(0.25)^2} = \frac{16}{17} \)- \( f(0.5) = \frac{1}{1+(0.5)^2} = \frac{4}{5} \)- \( f(0.75) = \frac{1}{1+(0.75)^2} = \frac{16}{25} \)- \( f(1) = \frac{1}{2} \)

Compute the Trapezoidal Sum

Use the trapezoidal rule formula: \[ \int_{0}^{1} \frac{d x}{1+x^{2}} \approx \frac{\Delta x}{2} \left( f(x_0) + 2f(x_1) + 2f(x_2) + 2f(x_3) + f(x_4) \right). \]Substitute the values: \[ \frac{1/4}{2} (1 + 2 \times \frac{16}{17} + 2 \times \frac{4}{5} + 2 \times \frac{16}{25} + \frac{1}{2}) = \frac{1}{8}(1 + \frac{32}{17} + \frac{8}{5} + \frac{32}{25} + \frac{1}{2}). \]

Evaluate the Expression to Approximate the Integral

Calculate the numerical value of the expression: - Convert the fractions to a common denominator and sum them up to approximately evaluate the integral. - Calculate this approximate value using the expression derived from the trapezoidal rule.

Use the Approximation to Estimate \(\pi\)

The exact result of the integral is \(\frac{\pi}{4}\). To approximate \(\pi\), multiply the result from Step 7 by 4 to estimate \(\pi\). Compare this result with the known value of \(\pi\).

Key concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are functions that reverse the action of their corresponding trigonometric functions. Among the key inverse trigonometric functions is the inverse tangent function (\( \tan^{-1}(x) \)). This function is notably important in calculus and appears frequently in integration problems.

For the given problem, the integrand \( \frac{1}{1+x^2} \) is recognized as the derivative of the inverse tangent function. This realization allows us to evaluate the integral analytically with ease. The relationship is framed by the antiderivative:

• \( \int \frac{d x}{1+x^2} = \tan^{-1}(x) + C \), where \( C \) is the constant of integration.

Using the limits of integration from 0 to 1, the Fundamental Theorem of Calculus simplifies the problem to:

• \( \tan^{-1}(1) - \tan^{-1}(0) \)

This equates to \( \frac{\pi}{4} \), illustrating the beautiful harmony between inverse trigonometric functions and definite integrals.

Thus, these functions play a crucial role in evaluating integrals that seem complex at first glance.

Trapezoidal Rule

The trapezoidal rule is a numerical method for approximating the value of definite integrals. It simplifies the integration process by approximating the region under the curve as a series of trapezoids, rather than curves.

To apply the trapezoidal rule, divide the interval of integration into equal subdivisions. In this case, the interval [0, 1] is divided into 4 equal parts (\( n=4 \)). Each part is then approximated as a trapezoid, rather than a segment of a curve. The rule is given by the formula:

• \( \int_{a}^{b} f(x) \, dx \approx \frac{\Delta x}{2} \left( f(x_0) + 2f(x_1) + 2f(x_2) + ... + 2f(x_{n-1}) + f(x_n) \right) \)

where \( \Delta x = \frac{b-a}{n} \) is the width of each trapezoid.

In our original problem, this approach allows us to turn what might be complex manual calculations into a computationally manageable process. By substituting the correct values into the formula and calculating, we arrive at an approximate value of the integral. Thus, the trapezoidal rule serves as a versatile tool when exact integration is challenging or when working equations by hand becomes cumbersome.

Approximation Methods

Approximation methods in calculus provide estimated solutions to problems that might be otherwise difficult to solve exactly. These methods offer a way to tackle definite integrals and other complex problems by providing a numerical value that is close to the exact answer.

Two major approximation methods include:

• Trapezoidal Rule - Uses trapezoids to approximate the area under the curve.
• Simpson's Rule - Uses parabolic arcs instead of straight trapezoidal lines, often improving accuracy when the function resembles parabolas.

In many practical scenarios, exact calculations are impossible or unnecessary, and approximations suffice to provide insights into the underlying mathematics.

By applying the trapezoidal rule to our original exercise, we not only approximated the integral of \( \frac{1}{1+x^2} \), but also used this result to estimate \( \pi \). This demonstrates the usefulness of approximation methods: they allow us to derive meaningful insights in applications ranging from engineering to physics, where precise analytical solutions may be hard to obtain. Overall, approximation tools simplify the complexities of calculus in practical and meaningful ways.