Problem 49 - Calculus: Early Transcendental Functions

Question

Explain why Simpson's Rule can't be used to approximate \(\int_{0}^{\pi} \frac{\sin x}{x} d x .\) Find \(L=\lim _{x \rightarrow 0} \frac{\sin x}{x}\) and argue that if \(f(x)=\left\\{\begin{array}{ll}\frac{\sin x}{x} & \text { if } x \neq 0 \\ L & \text { if } x=0\end{array} \text { then } \int_{0}^{\pi} f(x) d x=\int_{0}^{\pi} \frac{\sin x}{x} d x\right.\) Use an appropriate numerical method to conjecture that \(\int_{0}^{\pi} \frac{\sin x}{x} d x \approx 1.18\left(\frac{\pi}{2}\right)\)

Step-by-Step Solution

Answer

Simpson's Rule cannot be used directly to approximate \( \int_{0}^{\pi} \frac{\sin x}{x} dx \) because of the discontinuity at \( x = 0 \). However, by applying the limit and L'Hopital's rule, we can redefine the function such that it is continuous over the interval and thus apply Simpson's Rule. The approximate value of this integral, after applying Simpson's rule, is given to be \( 1.18\left(\frac{\pi}{2}\right) \).

Step 1: Analyze the Applicability of Simpson's Rule

Simpson's Rule can only be used for a function that is defined and smooth (i.e., continuous and differentiable) over the interval of integration. However, the given function \( \frac{\sin x}{x} \) has a point of discontinuity at \( x = 0 \). Therefore, Simpson's Rule cannot be used directly due to the discontinuity.

Step 2: Remove the Discontinuity and Determine L

The problematic point is \( x = 0 \). Here the given function takes the form \( \frac{0}{0} \), which is indeterminate. However, we can apply the limit and use L'Hopital's rule to find \( L = \lim_{x \to 0} \frac{\sin x}{x} \). After applying L'Hopital's rule, we get \( L = \lim_{x \to 0} \cos x \), which is 1 as cosine of 0 is 1. Thus, we can redefine the function over the interval of integration as \( f(x) = \left\{\begin{array}{ll}\frac{\sin x}{x} & \text { if } x \neq 0 \\ 1 & \text { if } x = 0\end{array}\right. \). This function is now continuous over the interval from 0 to \( \pi \), and therefore is suitable for Simpson's Rule.

Step 3: Approximate the Integral

Once the discontinuity has been removed, we can use an approximation method to estimate the integral \( \int_{0}^{\pi} f(x) d x \), which is approximately equal to \( 1.18\left(\frac{\pi}{2}\right) \) as the problem suggests.

Key Concepts

Numerical Methods

Numerical methods are mathematical strategies used to approximately solve problems when exact solutions are difficult or impossible to find.
These techniques are particularly useful in calculus for evaluating integrals that do not have closed-form solutions.
One of the commonly used numerical methods for approximating integrals is Simpson’s Rule.

• Simpson’s Rule is a technique that approximates the definite integral of a function by fitting parabolas under the curve.
• It’s most effective for smooth (continuous and differentiable) functions over the interval of integration.

Simpson’s rule is not the only numerical method available.

• Trapezoidal Rule is another approach that involves approximating the area under the curve using trapezoids instead of parabolas.
• Monte Carlo Integration uses randomness to estimate the value of an integral, which can be useful in higher dimensions.

These methods allow us to handle complex real-world problems, offering estimates where classical calculus might fall short.

Definite Integral

A definite integral can be thought of as the total accumulation of a quantity, such as area under a curve, over a specific interval on the x-axis.
It is a fundamental concept in calculus representing the signed area between the x-axis and the curve from one point to another.
Mathematically, a definite integral is denoted by: \[ \int_{a}^{b} f(x) \, dx \]

• The most crucial aspect of a definite integral is that it must be calculated over a continuous interval.
• This integral gives us a precise net area, useful in numerous fields like physics, engineering, and economics.

To evaluate integrals where the function isn’t easily integrable through analytical means, numerical methods such as Simpson's Rule can be employed, assuming continuity throughout the range of integration.

Discontinuous Functions

When dealing with functions in calculus, a discontinuity occurs where a function is not smooth, meaning it jumps, breaks, or otherwise does not have a continuous path.
For a function to be considered continuous at a certain point:

• The function must have a defined output at that point.
• The limit from both the left and right must exist and be equal to that output.

Discontinuities can be problematic when using methods like Simpson's Rule, as these methods assume a smooth path for approximations. However, functions with removable discontinuities, like in the case of \( rac{\sin x}{x} \) at \( x = 0 \), can often be managed by redefining the function at the point of indeterminacy to make it continuous. In this problem, by evaluating the limit \( L = \lim_{x \to 0} \frac{\sin x}{x} = 1 \),
we can redefine the function, making it amenable for numerical integration methods. Remember, ensuring continuity before applying numerical approximation is key to obtaining reliable estimates.