Problem 8 - Applied Calculus

Question

Approximate each integral using trapezoidal approximation "by hand" with the given value of \(n\). Round all calculations to three decimal places. \(\int_{0}^{1} e^{x^{2}} d x, \quad n=4\)

Step-by-Step Solution

Answer

The integral is approximately 1.491.

Step 1: Define the Integral and Subintervals

We are given the integral \( \int_{0}^{1} e^{x^2} \, dx \) with \( n = 4 \). This means that we need to divide the interval \([0, 1]\) into 4 equal subintervals. Since the interval length is \( 1 \), each subinterval will have a width of \( \Delta x = \frac{1 - 0}{4} = 0.25 \). Thus, the x-values at the subintervals are \( x_0 = 0, \ x_1 = 0.25, \ x_2 = 0.5, \ x_3 = 0.75, \ x_4 = 1 \).

Step 2: Evaluate the Function at Each Point

For each of these x-values, we need to evaluate the function \( e^{x^2} \) to use in the trapezoidal formula. We calculate: - \( f(x_0) = e^{0^2} = e^0 = 1.000 \)- \( f(x_1) = e^{(0.25)^2} \approx e^{0.0625} \approx 1.064 \)- \( f(x_2) = e^{(0.5)^2} = e^{0.25} \approx 1.284 \)- \( f(x_3) = e^{(0.75)^2} = e^{0.5625} \approx 1.756 \)- \( f(x_4) = e^{1^2} = e^1 = 2.718 \)Each of these values is rounded to three decimal places.

Step 3: Apply the Trapezoidal Rule

The trapezoidal approximation formula is:\[ T_n = \frac{\Delta x}{2} \left[ f(x_0) + 2\sum_{i=1}^{n-1} f(x_i) + f(x_n) \right] \]Substituting the computed values, we have:\[ T_4 = \frac{0.25}{2} \left[ 1.000 + 2(1.064 + 1.284 + 1.756) + 2.718 \right] \]\[ = 0.125 \left[ 1.000 + 2(4.104) + 2.718 \right] \]\[ = 0.125 \left[ 1.000 + 8.208 + 2.718 \right] \]\[ = 0.125 \times 11.926 \approx 1.491 \]

Step 4: Conclusion

By applying the trapezoidal rule, we approximate the integral \( \int_{0}^{1} e^{x^2} \, dx \) as approximately \( 1.491 \). This value is our final answer after rounding calculations to three decimal places.

Key Concepts

Numerical Integration

Numerical integration is a mathematical tool used for approximating the value of definite integrals. This process is especially useful when the integral cannot be solved analytically or the function is too complex. It involves techniques that estimate the area under a curve, providing an approximate value for the integral.

When performing numerical integration, various methods can be utilized, each with its own characteristics and levels of precision. The trapezoidal rule is one such method that is both straightforward and effective for many applications. It is especially helpful when dealing with functions that do not have elementary antiderivatives.

By applying numerical integration, calculations can be performed even when only discrete data points are available. This allows for wider use in practical fields such as physics, engineering, and finance.

Subintervals

To use the trapezoidal rule effectively, a large range is broken down into smaller, more manageable parts called subintervals. This is a crucial step in numerical integration. In our example, the interval \":[0, 1] is divided into 4 subintervals because the value of \(n\) is given as 4.

Breaking the interval into smaller pieces helps in increasing the accuracy of the approximation, as each segment assumes a linear approximation of the curve. The width of each subinterval is calculated as \(\Delta x = \frac{b-a}{n}\), where \(a\) and \(b\) are the bounds of the integral, and \(n\) is the number of subintervals.

For this specific problem, with \([0, 1]\) and \(n = 4\), each subinterval has a width of \(0.25\). The x-values marking the subinterval boundaries are important for evaluating the function at the specific points required by the trapezoidal rule.

Trapezoidal Rule

The trapezoidal rule is a simple yet powerful numerical technique to approximate the value of definite integrals. It replaces the area under the curve of a function with a series of trapezoids, whose areas are easy to compute.

This method computes the area by summing the areas of each trapezoid, which are determined by the function values at the endpoints of the subintervals. The formula used for the trapezoidal rule is:

• \( T_n = \frac{\Delta x}{2} \left[ f(x_0) + 2\sum_{i=1}^{n-1} f(x_i) + f(x_n) \right] \)

Here, \(x_0, x_1, \ldots, x_n\) are the points at which the function is evaluated, and \(\Delta x\) is the width of each subinterval.

The strength of the trapezoidal rule lies in its relatively low computational demands and its simplicity, making it a popular choice for quick estimates of integrals. However, for functions that are highly nonlinear, other methods might provide better accuracy.

Function Evaluation

Function evaluation is a critical step in numerical integration. It involves calculating the value of the function at specific points. For the trapezoidal rule, these points are the endpoints of each subinterval.

In the given problem, you need to evaluate the function \( e^{x^2} \) at each subinterval point: \( x_0 = 0, x_1 = 0.25, x_2 = 0.5, x_3 = 0.75, \) and \( x_4 = 1 \).

The calculations are as follows:

• \( f(x_0) = e^{0^2} = e^0 = 1.000 \)
• \( f(x_1) = e^{(0.25)^2} \approx e^{0.0625} \approx 1.064 \)
• \( f(x_2) = e^{(0.5)^2} = e^{0.25} \approx 1.284 \)
• \( f(x_3) = e^{(0.75)^2} = e^{0.5625} \approx 1.756 \)
• \( f(x_4) = e^{1^2} = e^1 = 2.718 \)

Each of these values is rounded to three decimal places for accuracy in the final trapezoidal computation. Proper function evaluation ensures that the trapezoidal rule is accurately applied, leading to a reliable approximation of the integral.