Problem 22 - 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 approximate value of the integral is 1.146.

Step 1: Set Up the Integral

The given integral to evaluate is \( \int_{1}^{4} \frac{1}{\sqrt{1+x^{2}}} \, dx \). We will perform integration by using a computational tool since it might involve complex functions.

Step 2: Identify the Function and Limits

We are integrating the function \( f(x) = \frac{1}{\sqrt{1+x^2}} \) over the interval from 1 to 4. This function does not have a simple antiderivative, so numerical methods or computational tools are recommended for evaluation.

Step 3: Use a Calculator or Computer

Utilize a calculator or a software tool such as Wolfram Alpha, Desmos, or a similar computational platform. Input the integral exactly as it appears: \( \int_{1}^{4} \frac{1}{\sqrt{1+x^{2}}} \, dx \).

Step 4: Evaluate the Integral

The computational tool will perform the integration, often using numerical methods, and provide the approximate result. Upon evaluation, the integral \( \int_{1}^{4} \frac{1}{\sqrt{1+x^{2}}} \, dx \) is approximately 1.146.

Key Concepts

Numerical Integration

When dealing with definite integrals that do not have a straightforward antiderivative, numerical integration steps in as a powerful technique. It allows us to approximate the area under complex curves, much like summing up slices of the curve's graph. Situations such as integrating \( \int_{1}^{4} \frac{1}{\sqrt{1+x^2}} \ dx \) present challenges for symbolic integration. That's where numerical methods shine.

• **Trapezoidal Rule**: Approximates the area using trapezoids.
• **Simpson’s Rule**: A more accurate method by applying parabolas.
• **Midpoint Rule**: Divides the curve into rectangles.

Each method involves partitioning the interval into small pieces. The smaller the partition, the more accurate the result. In practice, computational tools automate this process to save time and provide precision.

Computational Tools

Computational tools, such as calculators or computer software, are indispensable for modern numerical integration, especially for evaluating integrals with complex functions or at unusual intervals. They can quickly perform complicated calculations that would take considerable time if done manually. Popular tools for integration include:

• **Wolfram Alpha**
• **Desmos**
• **Graphing Calculators like TI-series**

These tools employ algorithms that handle both symbolic and numerical integration, adapting different methods according to the function's nature. For instance, inputting the integral \( \int_{1}^{4} \frac{1}{\sqrt{1+x^{2}}} \, dx \) in a computational tool would efficiently produce an approximate value using methods like Simpson's or adaptive quadrature.

Antiderivative

Antiderivatives or indefinite integrals are the reverse operation of differentiation. They help us find the original function whose derivative was given. In simple terms, if we differentiate an antiderivative, we end up with our original function. However, not every function has a simple antiderivative. The function \( \frac{1}{\sqrt{1+x^{2}}} \) is a good example of a function without a straightforward antiderivative. This arises due to the complexity and structure of the expression. For numerous functions, we rely on numerical or approximation methods because closed forms might not exist. Understanding when a function has a symbolic antiderivative can save a lot of time and effort, directing us when to use numerical integration right away.