Question

In Exercises \(49-54,\) use NINT to solve the problem. Evaluate \(\int_{-0.8}^{0.8} \frac{2 x^{4}-1}{x^{4}-1} d x\)

Short answer

The solution is a numerical value obtained after executing numerical integration of the function \(\frac{2x^4-1}{x^4-1}\) from \( -0.8 \) to \( 0.8 \).

Step-by-step solution

Understand the Problem

We are asked to evaluate a definite integral of a complex function which is \(\int_{-0.8}^{0.8} \frac{2 x^{4}-1}{x^{4}-1} dx\) using numerical integration. This problem does not have an elementary solution which is why we have to use numerical methods.

Setup for Numerical Integration

To set up numerical integration, we need to express the integral as a single value, or a sum of values, with distinct boundaries. In this case, the integral goes from \( -0.8 \) to \( 0.8 \) of the function \(\frac{2x^4-1}{x^4-1}\)

Execute the Numerical Integration

There are several methods of numerical integration such as Trapezoidal, Simpson's rule and so forth. The particular function mentioned in the problem can be complicated. It takes a software tool that can handle 'NINT' or numerical integration to perform this step. This typically involves dividing the area under the curve into multiple small areas and summing them up.

Interpreting the result

The result of the numerical integration is a numerical value which represents the definite integral of the function \(\frac{2x^4-1}{x^4-1}\) from \( -0.8 \) to \( 0.8 \). The interpretation of this value will depend on the context of the problem. In this case, since there is no other context, the value is the solution to our problem.

Key concepts

Definite Integral

The concept of a definite integral is fundamental to calculus. It represents the accumulation of quantities, such as area, volume, or mass, over a certain interval. Specifically, when we talk about a definite integral like \[\int_{a}^{b} f(x) dx\], we're referring to the sum of infinitesimally small areas under the curve of f(x) from point a to point b. Think of it as adding up an infinite number of \(f(x)\cdot dx\) slices between the bounds. In the given exercise, we are looking to find this accumulation for the function \(\frac{2 x^{4}-1}{x^{4}-1}\) between -0.8 and 0.8.

Understanding the definite integral helps us not only with simple geometric problems but also with more complex applications in engineering, physics, and economics, where we need to calculate things like work done by a force or the total income over a certain period.

Numerical Methods in Calculus

When functions are particularly complex or don't have elementary antiderivatives, we turn to numerical methods in calculus. These techniques are designed to approximate definite integrals, as exact answers may be impossible to find analytically. Numerical methods such as the Trapezoidal Rule, Simpson's Rule, and Monte Carlo Simulations convert the problem into a form where it is easier to compute an approximation of the integral. These methods essentially break down the area under the curve into shapes (like trapezoids or parabolic arcs) whose areas we can calculate.

Choosing between these methods often depends on the specific function you're dealing with and the level of accuracy required. For the problem in question, numerical integration is ideal since the function \(\frac{2x^4-1}{x^4-1}\) doesn't lend itself well to standard integration techniques.

NINT Function

The NINT function is a handy tool used in various mathematical software to perform numerical integration. It stands for 'Numerical INTegration' and is designed to compute the definite integral of a function when an analytical solution is not readily available.

Using NINT, the software divides the integration interval into small segments, calculates the area of each segment under the curve, and sums them to approximate the integral's value. This process is handled internally by the software, meaning that the user simply inputs the function and the limits of integration, and the NINT function takes care of the rest. This function is vital in our exercise as it allows us to compute the integral of \(\frac{2x^4-1}{x^4-1}\) from -0.8 to 0.8 without having to resort to complex calculations.

Integral Evaluation

Integral evaluation is the process of calculating the value of an integral, whether it be definite or indefinite. When we evaluate a definite integral using analytical methods, we find an antiderivative of the function and apply the Fundamental Theorem of Calculus by subtracting the value of this antiderivative at the lower limit from its value at the upper limit. However, when the function is too complex or when an antiderivative doesn't exist in terms of elementary functions, we must assess the integral numerically.

In the context of the given exercise, the evaluation of the integral \(\int_{-0.8}^{0.8} \frac{2 x^{4}-1}{x^{4}-1} dx\) is not straightforward analytically. This is where numerical evaluation steps in as an alternative, allowing us to approximate the value of the integral to a satisfactory degree of accuracy. Each method of numerical integration has its own level of precision and computational requirements, and choosing the appropriate one depends on the problem at hand.