Problem 69 - Applied Calculus

Question

Use a graphing calculator to evaluate each definite integral, rounding answers to three decimal places. [Hint: Use a command like FnInt or \(\left.\int \mathrm{f}(\mathrm{x}) \mathrm{d} \mathrm{x} .\right]\) $$ \int_{-1}^{1} e^{x^{2}} d x $$

Step-by-Step Solution

Answer

The integral evaluates to approximately 3.034.

Step 1: Understand the Definite Integral

This definite integral represents the area under the curve of the function \( e^{x^2} \) from \( x = -1 \) to \( x = 1 \). In this case, the function \( e^{x^2} \) is not easily integrable using elementary functions, so we will use a graphing calculator.

Step 2: Set Up the Integral in the Calculator

Turn on your graphing calculator and enter the math section where you can evaluate integrals. Look for a function such as "FnInt" or the symbol of an integral \( \int \). Here we are looking to evaluate \( \int_{-1}^{1} e^{x^2} \, dx \). Choose the option to input the lower limit (-1), the upper limit (1), and the function \( e^{x^2} \).

Step 3: Evaluate the Integral

Run the calculation on the graphing calculator, making sure it processes the function correctly between the specified limits. The calculator will output a numerical approximation of the area under the curve \( e^{x^2} \) from \( x = -1 \) to \( x = 1 \).

Step 4: Record the Result

Once the calculator has completed the computation, it will display a numerical result. Round this result to three decimal places to match the problem's requirements.

Key Concepts

Graphing Calculator

Graphing calculators are powerful tools used for performing complex mathematical calculations, including definite integrals, which are used to find area under curves. Using a graphing calculator to evaluate a definite integral, such as \( \int_{-1}^{1} e^{x^2} \, dx \), involves a few straightforward steps. First, you need to access the integral function on the calculator. This is typically found in the math menu and might be labeled as "FnInt" or with the integral symbol \( \int \).
Next, you input the function \( e^{x^2} \) and the limits of integration, which are \(-1\) and \(1\) in this case. The calculator then performs the numerical calculation to find the area under the curve of this function between these specified points.
Finally, the calculator provides the value, which represents the definite integral of the function over that range. Ensure to round this value to three decimal places, as required. This process is straightforward but requires careful input and understanding of what the calculator does.

Numerical Approximation

Numerical approximation is necessary when exact values are difficult or impossible to calculate analytically, as is the case with some definite integrals like \( \int_{-1}^{1} e^{x^2} \, dx \). A graphing calculator uses numerical methods to estimate these values.
These numerical methods, such as the Trapezoidal Rule or Simpson's Rule, break the area under a curve into smaller, manageable segments. They then sum the areas of these segments to provide an approximation of the integral. The calculator quickly performs these complex calculations, offering a reliable estimate of the integral's value.

• Provides an easier method for complex integrals
• Ensures precise approximations
• Allows for results rounded to desired decimal places

In the context of using a graphing calculator, numerical approximation simplifies complex calculations, giving students a practical means to handle difficult functions like \( e^{x^2} \). This approach is valuable in both educational settings and real-world applications where exact solutions are not feasible.

Exponential Functions

Exponential functions, such as \( e^{x^2} \), can present challenges when trying to solve definite integrals analytically. These functions are characterized by a constant base raised to a power that includes a variable exponent.
One main property of exponential functions is their rapid growth or decay, depending on the sign of the exponent. In the integral \( \int_{-1}^{1} e^{x^2} \, dx \), the function \( e^{x^2} \) grows as \( x \) increases, making the area under the curve more complex to compute without a calculator.

• The function \( e^{x^2} \) does not have a simple antiderivative in terms of elementary functions.
• Requires numerical methods or technology for integration.
• Characterized by rapid changes in growth rate.

Understanding the nature of exponential functions helps in grasping why specific methods, like numerical approximation, are necessary.
This knowledge is essential for tackling real-world problems involving growth processes and other phenomena modeled by exponential functions. It enhances comprehension when utilizing tools like graphing calculators for complex calculations.