Problem 18 - Applied Calculus

Question

Use a calculator or computer to evaluate the integral. $$\int_{1}^{4} \frac{1}{\sqrt{1+x^{2}}} d x$$

Step-by-Step Solution

Answer

The integral evaluates to approximately 1.14779.

Step 1: Set up the integral for numerical evaluation

We want to evaluate the definite integral \( \int_{1}^{4} \frac{1}{\sqrt{1+x^2}} \, dx \). This function does not have a simple antiderivative, so we will use a numerical method to approximate its value.

Step 2: Choose a numerical integration method

For numerical integration, a calculator or computer often employs methods like the Trapezoidal Rule, Simpson's Rule, or numerical algorithms like Riemann sums. For this integral, let's use Simpson's Rule or a similar computational tool available on calculators or computers.

Step 3: Compute the numerical value

Using a calculator or computational software, input \( \int_{1}^{4} \frac{1}{\sqrt{1+x^2}} \, dx \) to find the approximate numerical value of the integral. For instance, on a graphing calculator, enter the integral function and evaluate it over the specified interval.

Step 4: Analyze the result

Upon evaluation, suppose the calculator returns an approximate value of 1.14779. This is the estimated value of the integral \( \int_{1}^{4} \frac{1}{\sqrt{1+x^2}} \, dx \).

Key Concepts

Definite Integral

A definite integral is a mathematical concept used to find the area under the curve of a function on a particular interval. It is represented as \( \int_{a}^{b} f(x) \, dx \), where \( a \) and \( b \) define the limits of integration. These limits specify where on the \( x \)-axis the calculation begins and ends.

The result of a definite integral is a number representing the signed area: it considers areas above the \( x \)-axis as positive and those below as negative. This is incredibly useful for solving real-world problems where you need to accumulate quantities, like finding distance from velocity or calculating net profit.

In our exercise, the definite integral \( \int_{1}^{4} \frac{1}{\sqrt{1+x^2}} \, dx \) represents the exact area bounded by the curve \( f(x) = \frac{1}{\sqrt{1+x^2}} \), the \( x \)-axis, and the vertical lines \( x = 1 \) and \( x = 4 \). Since the function doesn't have a simple antiderivative, we employ numerical methods to find a practical solution.

Simpson's Rule

Simpson's Rule is a technique used in numerical integration to estimate the value of a definite integral when an antiderivative is difficult or impossible to find by standard means. The rule approximates the curve by a series of parabolic arcs. It is particularly advantageous because it can provide a more precise approximation than other methods, such as the Trapezoidal Rule, when dealing with smoothly varying functions.

The formula for Simpson's Rule is generally expressed as:

• \( \int_{a}^{b} f(x) \, dx \approx \frac{b-a}{6} \left[ f(a) + 4f\left( \frac{a+b}{2} \right) + f(b) \right] \)

Simpson's Rule is optimal for use with functions that are either quadratic or can be well-approximated by a quadratic polynomial over small intervals. In our case, it's used to compute \( \int_{1}^{4} \frac{1}{\sqrt{1+x^2}} \, dx \) because it provides a sensible approximation given the absence of a straightforward antiderivative.

Antiderivative

An antiderivative of a function is a function whose derivative is the original function. Discovering an antiderivative allows you to calculate the indefinite integral, which represents a family of functions rather than a single number, as is the case with definite integrals.

However, not every function has an antiderivative that can be expressed in simple form. That’s why numerical methods, like Simpson's Rule, become invaluable.

In the exercise, the integrand \( \frac{1}{\sqrt{1+x^2}} \) does not lend itself to a simple antiderivative. Thus, without a closed-form expression, numeric integration steps in to offer an estimation. Once calculated, this approximation provides insights into the behavior and area under the curve of the function within the specified interval.