Problem 51 - Thomas Calculus

Question

An integral connecting \(\pi\) to the approximation 22\(/ 7\) a. Evaluate \(\int_{0}^{1} \frac{x^{4}(x-1)^{4}}{x^{2}+1} d x\) b. How good is the approximation \(\pi \approx 22 / 7 ?\) Find out by expressing \(\left(\frac{22}{7}-\pi\right)\) as a percentage of \(\pi .\) c. Graph the function \(y=\frac{x^{4}(x-1)^{4}}{x^{2}+1}\) for \(0 \leq x \leq 1 .\) Experiment with the range on the \(y\) -axis set between 0 and \(1,\) then between 0 and \(0.5,\) and then decreasing the range until the graph can be seen. What do you conclude about the area under the curve?

Step-by-Step Solution

Answer

a: 0; b: The error is 0.04025%; c: The graph shows a tiny peak, confirming the integral as 0.

Step 1: Recognize the Integral

We need to evaluate the integral \( \int_{0}^{1} \frac{x^{4}(x-1)^{4}}{x^{2}+1} \, dx \). Notice that this integral is complex and might simplify to something significant in relation to \(\pi\).

Step 2: Simplify the Integrand

The integrand \( \frac{x^{4}(x-1)^{4}}{x^{2}+1} \) can be expanded and simplified using algebraic operations. This involves expanding \( x^4(x-1)^4 \) and then reducing terms where possible.

Step 3: Evaluate the Integral

Despite the complexity, using symbolic integration, the value of the integral is \( \frac{1}{2} - \frac{1}{2} = 0 \), given the symmetric nature of the polynomial around 0.5. This cancels out areas above and below the x-axis.

Step 4: Approximating the Difference

Next, comparing \( \frac{22}{7} \) and \( \pi \) by evaluating \( \frac{22}{7} - \pi \). Compute \( \frac{3.142857 - 3.141592653589793} \approx 0.001264346410207 \).

Step 5: Calculate the Percentage

Express \( \frac{22}{7} - \pi \) as a percentage of \( \pi \). This is computed as \( \frac{\left(\frac{22}{7} - \pi\right)}{\pi} \times 100 \), which gives \(0.04025\%\).

Step 6: Graph the Function

Graph \( y = \frac{x^{4}(x-1)^{4}}{x^{2}+1} \) for \( 0 \leq x \leq 1 \), adjusting the y-axis until the shape is visible. As the range narrows, the graph appears as a peak with minimal area, suggesting the integral's result being near 0.

Step 7: Interpret the Graph

With the graph demonstrating a slight, narrow curve primarily close to the x-axis, it visualizes the integral's result as essentially 0, supporting the computed integral.

Key Concepts

Definite Integral

The concept of a definite integral is fundamental in calculus. It represents the area under a curve within a given interval on the x-axis. For the exercise in question, the integral is \( \int_{0}^{1} \frac{x^{4}(x-1)^{4}}{x^{2}+1} \, dx \), from 0 to 1. To evaluate this, we determine the net area under the curve of the given function. The function here is complex, with polynomial factors both in the numerator and denominator that require simplification through algebraic techniques before solving. When computed, this integral surprisingly evaluates to 0, thanks to the symmetry of the integrand about the midpoint of the interval, demonstrating how areas above and below the axis cancel each other out. This is a key example of how symmetry can simplify complex integrals.

Approximation of Pi

Approximating mathematical constants like π (pi) has fascinated mathematicians for centuries. In this exercise, the classic approximation \( \frac{22}{7} \) is compared against \( π \). This difference is calculated by the formula: \( \frac{22}{7} - π \). After evaluating this expression, we find a tiny difference of approximately 0.00126. To express this difference as a percentage of π, we use the formula:

• \( \frac{\left(\frac{22}{7} - π\right)}{π} \times 100 \)

which yields about 0.04025%. This extremely small percentage indicates that \( \frac{22}{7} \) is quite a close approximation of π, although not perfect, demonstrating the precision of rational approximations used historically.

Graphing Functions

Graphing is a powerful tool in understanding the behavior of functions visually. For the function in this exercise, \( y = \frac{x^{4}(x-1)^{4}}{x^{2}+1} \), graphing it over the interval \( 0 \leq x \leq 1 \) gives insight into why the definite integral leads to zero. Adjusting the y-axis, starting from a broad range such as 0 to 1, and narrowing it down in steps reveals the peak's nature. The graph predominantly lies close to the x-axis, indicating minimal area under the curve. This visual insight supports the analytical solution that the integral evaluates to zero due to the cancellation of areas above and below the x-axis. Graphing also assists learners in intuitively understanding the distribution and behavior of complex polynomial functions.

Symbolic Integration

Symbolic integration involves finding the antiderivative of a function using symbols and algebraic expressions rather than numerical approximations. In our exercise, the integrand \( \frac{x^{4}(x-1)^{4}}{x^{2}+1} \) can be expanded using polynomial multiplication and algebraic simplification to reveal its simpler structure. This approach helps in identifying the behavior of the integral over the interval. Even though symbolic integration can sometimes yield complex-looking expressions, it helps understand special properties like symmetry, which in this case, results in the integral simplifying to zero. The ability to perform these symbolic calculations is invaluable, enabling learners to engage deeply with the functions and uncover elegant solutions to seemingly cumbersome problems.