Question

Approximate the integral to three decimal places using the indicated rule. \(\int_{0}^{3} \frac{1}{1+x^{3}} d x ;\) Simpson's rule; \(n=3\)

Short answer

The approximate integral is 0.785.

Step-by-step solution

Understand Simpson's Rule

Simpson's Rule is a numerical method to approximate the value of a definite integral. It is given by the formula: \[\int_{a}^{b} f(x) dx \approx \frac{\Delta x}{3} [f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + f(x_4)]\]where \( \Delta x = \frac{b-a}{n} \) and \( n \) is even.

Calculate \( \Delta x \)

Given \( a = 0 \), \( b = 3 \), and \( n = 3 \). First, we need \( n \) to be even, so use \( n = 4 \). Now calculate \( \Delta x = \frac{b-a}{n} = \frac{3-0}{4} = 0.75 \).

Identify \( x \, \text{values} \)

Using \( \Delta x = 0.75 \), calculate \( x \) values: \[ x_0 = 0, \ x_1 = 0.75, \ x_2 = 1.5, \ x_3 = 2.25, \ x_4 = 3 \]

Evaluate the Function at Each \( x \, \text{value} \)

Compute \( f(x) = \frac{1}{1+x^3} \) at each \( x \):\[ f(0) = 1, \ f(0.75) = \frac{1}{1+0.75^3}, \ f(1.5) = \frac{1}{1+1.5^3}, \ f(2.25) = \frac{1}{1+2.25^3}, \ f(3) = \frac{1}{1+3^3} \]

Apply Simpson's Rule

With the values, apply Simpson's rule:\[\int_{0}^{3} \frac{1}{1+x^{3}} dx \approx \frac{0.75}{3}[1 + 4f(0.75) + 2f(1.5) + 4f(2.25) + \frac{1}{28}]\]Calculate each value and sum according to the formula.

Calculate Numeric Integral

Substitute the function values into the formula and perform the arithmetic:\[\int_{0}^{3} \frac{1}{1+x^{3}} dx \approx \frac{0.75}{3} \times (1 + 4 \times 0.8219178 + 2 \times 0.296296 + 4 \times 0.081945 + 0.035714)\]\[\approx \frac{0.75}{3} \times 3.139276 \approx 0.7848\]Then round to three decimal places.

Key concepts

Numerical Integration

Numerical integration is a method used to approximate the value of definite integrals, particularly when an integral cannot be solved analytically. It is especially useful for functions that do not have elementary antiderivatives or when precision is not critical, and a numerical solution suffices. There are several techniques to perform numerical integration, with Simpson's Rule being one of them.
Simpson's Rule is known for its accuracy and efficiency. It uses parabolas to approximate the actual curve of the function over the interval of integration, leading to more precise results compared to other methods such as the midpoint or trapezoidal rule.
Key features of numerical integration include:

• Approximating complex or unsolvable integrals.
• Handling functions difficult to integrate analytically.
• Providing simplicity and efficiency in computation.

These features make numerical methods an essential tool in fields like engineering, physics, and finance where practical solutions are needed.

Definite Integral

A definite integral is a mathematical construct, representing the accumulation of quantities, such as area under a curve, over a specified interval. Mathematically, it is represented as \( \int_{a}^{b} f(x) \, dx \), where \( a \) and \( b \) denote the lower and upper bounds of integration, respectively.
In the context of Simpson's Rule, the definite integral of a function \( f(x) \) over the interval \([a, b]\) is approximated by summing the areas under parabolic segments that fit the curve. The calculus of these segments leads to a sum of weighted function values at specific points or nodes determined by the rule.
Notably, the definite integral provides:

• The total accumulation of value, such as distance or area, over an interval.
• A means to evaluate physical quantities in real-world applications, like calculating probabilities or quantities of materials.
• Insights into the behavior of functions, helping to understand rates of change and accumulation.

Understanding definite integrals is crucial for solving real-world problems involving continuous data and evaluating overall trends.

Delta x Calculation

In numerical integration, particularly when applying Simpson's Rule, calculating \( \Delta x \) is fundamental to dividing the interval of integration into even subintervals. The formula \( \Delta x = \frac{b-a}{n} \) helps determine the width of each subinterval.
Importantly, for Simpson's Rule, \( n \), the number of subintervals, must be even to ensure that each complete interval includes an endpoint. If presented with an odd \( n \), it is traditionally adjusted to the next even integer to satisfy the method's requirements.
Here's a quick guide to the \( \Delta x \) calculation:

• Identify the limits of integration \( a \) and \( b \).
• Choose or adjust \( n \) to be an even number for optimal use of Simpson's Rule.
• Calculate the subinterval width using \( \Delta x = \frac{b-a}{n} \).

Remember, getting \( \Delta x \) correct ensures an accurate placement of node points for evaluating the function, crucial for achieving an accurate approximation with Simpson's Rule.