Question

In Exercises 51-54, use a power series to approximate the value of the integral with an error of less than \(0.0001 .\) (In Exercises 51 and \(52,\) assume that the integrand is defined as 1 when \(x=0 .)\) $$ \int_{0}^{1} \frac{\sin x}{x} d x $$

Short answer

The integral is approximately equal to 0.946083, computed as the sum of the first 4 terms of the series \(1 - \frac{1}{3 \cdot 3!} + \frac{1}{5 \cdot 5!} - \frac{1}{7 \cdot 7!}\).

Step-by-step solution

Remove the singularity

Firstly, recall that although \(\frac{\sin x}{x}\) is undefined at \(x = 0\), the limit of \(\frac{\sin x}{x}\) as \(x\) tends to 0 is 1. Therefore, assume that \(\frac{\sin x}{x} = 1\) when \(x = 0\). This will remove the point of singularity.

Finding a power series for \(\frac{\sin x}{x}\)

To proceed with this, recall the Taylor series expansion of the sine function, i.e., \(\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \ldots\). Dividing both sides by \(x\) yields: \(\frac{\sin x}{x} = 1 - \frac{x^2}{3!} + \frac{x^4}{5!} - \frac{x^6}{7!} + \ldots\). This is a valid power series representation for \(\frac{\sin x}{x}\) on the interval of integration minus the point \(x = 0\). However, as we've already removed the singularity in the previous step, it's also valid at this point.

Compute the integral

Now, use the power series representation to compute the integral approximately by integrating each term of the series individually on the interval [0,1]: \(\int_{0}^{1}\left(1 - \frac{x^2}{3!} + \frac{x^4}{5!} - \frac{x^6}{7!} + \ldots\right) dx = 1 - \frac{1}{3 \cdot 3!} + \frac{1}{5 \cdot 5!} - \ldots\).

Compute the series to the desired accuracy

Finally, as the original series converged absolutely on [0,1] - due to the fact that sin x/x is a bounded function on this interval -, the above series also converges absolutely. Therefore, compute the first few terms of this series such that the difference between the last computed term and the next one is less than \(0.0001\). This will ensure that the error of the whole computation is less than \(0.0001\) too. After such computation, you can get an approximation for the integral.

Key concepts

Taylor Series Expansion

A Taylor series expansion allows mathematicians and scientists to approximate complex functions with a series of polynomial terms that are derived from the function's derivatives at a single point. At the heart of this method is the idea that any smooth function can be represented as an infinite sum of its derivatives. The formula for the Taylor series of a function f(x) around the point a is given by:
\[ f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + \ldots \]
This mathematical tool is incredibly useful when it's difficult to work directly with the original function. In our problem, we used a Taylor series to find a simpler representation of the sine function, which we then used to integrate and approximate a value with a small error margin. Importantly, one must determine the number of terms needed to achieve the desired level of accuracy—in this case, less than 0.0001.

Integral Approximation

Integral approximation is the process of calculating the value of an integral using methods that yield a value close to, but not necessarily equal to, the exact solution. A common and valuable tool for approximating integrals is utilizing series expansions, such as the Taylor series mentioned earlier. For the given exercise, after expanding the sine function into its power series form, we calculate the integral by taking the integral of each term in the power series.
By integrating each term of the power series representation of \( \frac{\sin x}{x} \), we get a series of constants, each of which corresponds to the integral of a term in the power series. We keep adding these constants until the difference between consecutive terms is less than 0.0001, securing the desired level of precision. Achieving such a threshold ensures that the integral's approximation aligns with the stated error margin, making our approach both practical and reliable for estimating the value of the otherwise complex integral.

Sine Function

The sine function, represented as \( \sin(x) \), is a fundamental trigonometric function that describes the ratio of the length of the side of the triangle opposite the angle to the length of the triangle's hypotenuse in a right-angled triangle. However, its importance extends far beyond geometry; in calculus, the sine function is central to modeling periodic phenomena like waves.
In the context of series, particularly the Taylor series, the sine function can be expressed as an infinite polynomial that beautifully captures its oscillatory nature:
\( \sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \ldots \)
The function's smooth and periodic characteristics assure the convergence of its power series, which is critical when using this series for calculations like integral approximation, as was done to solve the textbook exercise. By using a power series to approximate \( \sin x \), we are able to perform complex integrations on functions with this component, which include many real-world applications in physics and engineering.