Question

Approximate the integral to three decimal places using the indicated rule. \(\int_{0}^{0.8} e^{-x^{2}} d x ;\) trapezoidal rule \(; n=4\)

Short answer

Approximately 0.655

Step-by-step solution

Understand the Trapezoidal Rule Formula

The trapezoidal rule is used to approximate the definite integral of a function. For a function \(f(x)\) over the interval \([a, b]\) divided into \(n\) subintervals, the approximation is given by: \[ T_n = \frac{\Delta x}{2} \left( f(x_0) + 2f(x_1) + 2f(x_2) + \cdots + 2f(x_{n-1}) + f(x_n) \right) \]where \(\Delta x = \frac{b-a}{n}\) and \(x_i = a + i \Delta x\) for \(i = 0, 1, 2, \ldots, n\).

Calculate the Width of Each Subinterval

Given that \(a = 0\), \(b = 0.8\), and \(n = 4\), we calculate \(\Delta x\) as follows:\[ \Delta x = \frac{0.8 - 0}{4} = 0.2 \]

Determine the Subinterval Points

We calculate the x-values for the trapezoidal rule as follows:- \(x_0 = 0\)- \(x_1 = 0.2\)- \(x_2 = 0.4\)- \(x_3 = 0.6\)- \(x_4 = 0.8\)

Evaluate the Function at Each Subinterval Point

Compute the values of \(f(x) = e^{-x^2}\) at each point:- \(f(x_0) = e^{-(0)^2} = e^{0} = 1\)- \(f(x_1) = e^{-(0.2)^2} \approx 0.961\)- \(f(x_2) = e^{-(0.4)^2} \approx 0.852\)- \(f(x_3) = e^{-(0.6)^2} \approx 0.697\)- \(f(x_4) = e^{-(0.8)^2} \approx 0.527\)

Apply the Trapezoidal Formula

Substitute the function values into the trapezoidal rule formula:\[ T_4 = \frac{0.2}{2} \left( 1 + 2(0.961) + 2(0.852) + 2(0.697) + 0.527 \right) \]Calculate:\[ T_4 = 0.1 (1 + 1.922 + 1.704 + 1.394 + 0.527) \]\[ T_4 = 0.1 \times 6.547 \]\[ T_4 = 0.6547 \]

Approximate the Integral

The approximate value of the definite integral \( \int_{0}^{0.8} e^{-x^{2}} \, dx \) using the trapezoidal rule with \(n = 4\) is \(0.655\) to three decimal places.

Key concepts

Numerical Integration

Numerical integration is a computational technique used to find approximate solutions to definite integrals, especially when an exact solution is difficult or impossible to calculate analytically. This process is carried out by applying various mathematical formulas and methods that estimate the area under the curve of a function. Given that traditional calculus-based integration might not be feasible for complex functions, numerical integration offers a practical way to tackle these problems.

• It deals with approximating the value of integrals, which represents accumulated change or area under a curve.
• Common methods include the trapezoidal rule, Simpson's rule, and midpoint rule, each with its own level of accuracy and computational complexity.
• Numerical integration is widely used in fields such as physics, engineering, and finance where integral values are essential for making predictions and informed decisions.

Overall, numerical integration is a versatile and essential tool for approximating integrals when analytical solutions are not available.

Approximate Integral

An approximate integral provides an estimation of the integral value rather than an exact answer. This is particularly helpful when dealing with functions that are difficult or impossible to integrate analytically. Approximations can still provide valuable insights into the behavior of the function over a certain interval.
To estimate integrals, we harness numerical methods that transform the problem into a more manageable form. These methods convert the continuous function into discrete segments or partitions, making it easier to sum up the contribution of each segment.

• Approximation serves as a pragmatic approach to gaining insights into complex functions.
• The accuracy of the approximation is influenced by factors such as the number of subintervals used and the method applied.

Using techniques like the trapezoidal rule, students can efficiently approximate integrals with a balance between computational effort and accuracy.

Definite Integral

A definite integral represents the accumulation of quantities, such as area under a curve, over a specified interval \([a, b]\). Calculating a definite integral amounts to finding the total net change or total area between the curve and the x-axis over the given range.

• The process involves finding the antiderivative of the function that sums up the infinitesimal contributions along the interval.
• Definite integrals are bounded and yield a numerical value, as opposed to indefinite integrals which provide a family of functions.
• Definite integrals have real-world applications in physics to calculate quantities such as work, energy, and probability.

In numerical integration, approximate solutions are calculated to emulate the value of the related definite integral, facilitating insights into real-world scenarios where exact solutions are elusive.

Subintervals

Subintervals are smaller divisions of the overall interval \([a, b]\) over which integration occurs. These are essential for numerical integration techniques, as they allow the integral to be approximated through summing individual contributions from each subinterval.In the trapezoidal rule, the chosen interval is divided into equal subintervals. Each subinterval thus represents a slice of the total area under the function's curve, which we approximate.

• The interval width \( \Delta x \) is obtained by dividing the difference between the endpoints \( b - a \) by the number of subintervals \( n \).
• The larger the number of subintervals, the more accurate the approximation of the integral, as more points are captured along the curve.
• Key to effective numerical integration is selecting a suitable number of subintervals that balances accuracy and computational workload.

This division and approximation allow for a manageable calculation and a nuanced estimate of the integral value.