Problem 29 - Calculus Early Transcendentals: Pearson New International Edition

Question

Calculate, accurate to four decimal places, $$ \int_{0}^{1} \cos \left(x^{2}\right) d x $$

Step-by-Step Solution

Answer

\( \int_{0}^{1} \cos(x^2) \approx 0.7408 \)

Step 1: Understanding the Problem

We need to evaluate the definite integral \( \int_{0}^{1} \cos(x^2) \, dx \). This is a definite integral from \( x = 0 \) to \( x = 1 \) of the function \( \cos(x^2) \).

Step 2: Recognizing the Method Required

The integrand \( \cos(x^2) \) does not have an elementary antiderivative. Therefore, to find the value of this integral, we must resort to numerical methods.

Step 3: Choosing Integration Technique

One common numerical method is Simpson's Rule, which is suitable for providing results accurate to four decimal places. Simpson's Rule approximates the integral by dividing the area under the curve into parabolic segments.

Step 4: Applying Simpson's Rule

For Simpson's Rule, with \( n = 2 \), let \( h = \frac{b-a}{n} = 0.5 \), and calculate the points \( x_0 = 0 \), \( x_1 = 0.5 \), and \( x_2 = 1 \). Then compute: \[ I \approx \frac{h}{3} \left[ \cos(0^2) + 4\cos(0.5^2) + \cos(1^2) \right] = \frac{0.5}{3} [1 + 4\cos(0.25) + \cos(1)] \]

Step 5: Performing the Calculations

Calculate each cosine value: \( \cos(0.25) \approx 0.9689 \), \( \cos(1) \approx 0.5403 \). Substitute back: \[ I \approx \frac{0.5}{3} (1 + 4 \times 0.9689 + 0.5403) \approx \frac{0.5}{3} (4.4456) \approx 0.7408 \]

Step 6: Verifying the Solution

The approximate integral value, using Simpson's Rule with \( n = 2 \), comes out to approximately 0.7408, which matches typical results for this integral using numerical methods.

Key Concepts

Simpson's Rule

When it comes to numerical integration, Simpson's Rule stands out as a pretty popular choice. This method gives a way to estimate the definite integral of a function, especially when finding the antiderivative seems tricky or impossible. It works by approximating the area under the curve by breaking it into sections shaped like parabolas.

Here's a quick rundown of how it works:

• Divide the interval of integration into an even number of subintervals.
• Each pair of subintervals forms a segment that approximates the function with a parabola.
• The area under these parabolas is calculated, which gives a good estimate of the integral.

The accuracy of Simpson's Rule can be quite impressive, especially when compared to basic strategies like the Trapezoidal Rule. It's great for functions that aren't easy to integrate using standard methods, like our function here, \( \cos(x^2) \).

Definite Integrals

A definite integral, in a nutshell, is a way to sum up infinite points on a curve to find the area under it, defined over a specific interval. For instance, the integral \( \int_{0}^{1} \cos(x^2) \; dx \) seeks the area from \( x = 0 \) to \( x = 1 \) for the function \( \cos(x^2) \).

These integrals are definite because they are bound by finite limits. Unlike indefinite integrals that yield a family of functions, definite integrals provide an actual numerical value, representing the signed area between the function and the x-axis within the specified interval. This is crucial in real-world applications where numerical results are often needed.

Numerical Methods

Numerical methods are a group of techniques often deployed when exact solutions to mathematical problems are not feasible. With calculus, this often means finding approximate solutions to integrals that don't have simple antiderivatives. In our case, we used Simpson's Rule to approximate the integral of \( \cos(x^2) \).

There are various approaches within numerical integration:

• Trapezoidal Rule - uses trapezoids to approximate the area.
• Midpoint Rule - uses rectangles centered around midpoints.
• Simpson's Rule - as discussed, uses parabolic arcs.
• Gaussian Quadrature - an advanced method for higher accuracy.

Each method comes with its use cases, advantages, and levels of accuracy. Understanding how they work can help choose the right one for your problem.

Cosine Function

The cosine function, denoted as \( \cos(x) \), is a periodic function fundamental in the study of trigonometry. It is the x-coordinate of a point on the unit circle as you rotate it through an angle \( x \).

Key features of the cosine function include:

• It oscillates between -1 and 1.
• Has a period of \( 2\pi \).
• Symmetric about the y-axis, making it an even function.

In calculus, the cosine function plays a significant role, particularly because some of its variations, like \( \cos(x^2) \), don't lend themselves to an easy antiderivative. This requires numerical methods, like what we are exploring, to evaluate integrals.

Accuracy in Calculus

Achieving high accuracy in calculus is essential, especially when results inform critical decisions in engineering or sciences. Numerical integration methods are practical tools, but understanding their accuracy limits is crucial.

Simpson's Rule, for example, provides good accuracy, typically within a few decimal places if chosen appropriately for the function and interval of interest. However, it depends on factors such as:

• The smoothness of the function
• The number of subintervals
• The choice of method itself

Understanding these factors helps ensure that when we apply methods like Simpson's Rule, we get results that are not only approximate but trustably close to the true integral. In our calculation, we've reached an accuracy within four decimal places, which is generally considered quite precise for most applications.