Question

Approximate the integral using (a) the Trapezoidal Rule and (b) Simpson's Rule for the indicated value of \(n\). (Round your answers to three significant digits.) $$ \int_{0}^{2} \frac{1}{\sqrt{1+x^{3}}} d x, n=4 $$

Short answer

After performing all the necessary calculations, round your answer to three significant digits. It is important to remember that these are approximations, hence the actual answer might be slightly different.

Step-by-step solution

Using the Trapezoidal Rule

The formula for the Trapezoidal Rule is \( T_{n} = \frac{b-a}{2 n} [f(a) + 2 \sum_{i=1}^{n-1} f(a + i \cdot \frac{b-a}{n}) + f(b)]\), where \( f(x)\) is the function, \( a\) is the lower limit, \( b\) is the upper limit, and \( n\) is the number of trapezoids. In this case, \( f(x) = \frac{1}{\sqrt{1+x^{3}}}\), \( a = 0\), \( b = 2\), and \( n = 4\). It requires plugging these values into the formula, calculating the inner sum, and executing the outer operations to obtain the approximation.

Using Simpson's Rule

The formula for Simpson's Rule is \( S_{n} = \frac{b - a}{3n} [f(a) +4\sum_{i=1}^{2n-1} f(a+i\cdot \frac{b - a}{2n}) +2\sum_{i=2}^{2n-2} f(a+i\cdot \frac{b - a}{2n})+ f(b)] \), where \( f(x)\), \( a\), \( b\), and \( n\) are defined as before. It requires plugging these values into the formula, calculating the two inner sums, and executing the outer operations to obtain the approximation.

Key concepts

Trapezoidal Rule

The Trapezoidal Rule is a method of numerical integration used to approximate the value of a definite integral. Instead of calculating the area under a curve using an exact method, we use trapezoids to estimate the space beneath the curve. This is especially useful when dealing with complex integrals that cannot be solved easily by conventional means.

To apply the Trapezoidal Rule, we divide the interval \[a, b\] into \(n\) subintervals. For each subinterval, we approximate the area under the curve by using a trapezoid. The area of each trapezoid is calculated and then summed to get the total approximation for the entire integral.

The formula for the Trapezoidal Rule is given by:
\[T_{n} = \frac{b-a}{2n} \left[f(a) + 2 \sum_{i=1}^{n-1} f(a + i \cdot \frac{b-a}{n}) + f(b) \right]\]

### Steps to Apply the Trapezoidal Rule:
- Identify the function \(f(x)\) and the limits of integration \(a\) and \(b\).
- Decide the number of subintervals \(n\).
- Compute the width of each subinterval \(\Delta x = \frac{b-a}{n}\).
- Calculate \(f(x)\) at each point and plug these into the trapezoidal formula.
- Finally, compute the sum to find the approximate value of the integral.

The accuracy of this method increases with the number of trapezoids (\(n\)), but it also depends on how smooth the function is within the interval. Though simple, it is a powerful and frequently used method in approximation calculus.

Simpson's Rule

Simpson's Rule is another widely used numerical integration technique that provides a more accurate estimate of the definite integral compared to the Trapezoidal Rule.

Simpson’s Rule goes a step further by approximating the portions under the curve using parabolic arcs rather than straight lines. This method works especially well when the curve is smooth, making the result even more precise than approximations using trapezoids.

The formula for Simpson’s Rule is:
\[S_{n} = \frac{b - a}{3n} \left[f(a) + 4\sum_{i=1}^{2n-1} f(a+i\cdot \frac{b - a}{2n}) + 2\sum_{i=2}^{2n-2} f(a+i\cdot \frac{b - a}{2n})+ f(b)\right]\]

### Applying Simpson's Rule:
- Determine the function \(f(x)\) and define limits \(a\) and \(b\).
- Select an even number of subintervals \(n\). (Simpson’s Rule requires an even number for \(n\).)
- Determine the step size \(\Delta x = \frac{b-a}{2n}\).
- Calculate \(f(x)\) at each necessary point and apply it into Simpson’s formula.
- Calculate the result to obtain the integral's approximation.

Simpson’s Rule is often more accurate than the Trapezoidal Rule, particularly when the actual function closely resembles a quadratic equation within the considered interval. This increased accuracy comes from the method’s use of parabolas for integration, making it a choice technique in scenarios demanding high precision.

Approximation Methods

Approximation methods, such as the Trapezoidal Rule and Simpson's Rule, are critical when dealing with integrals that are difficult or impossible to solve analytically. These methods offer a way to estimate the integral's value without requiring exact calculations.

### Overview of Approximation Methods
- **Trapezoidal Rule**: Uses trapezoids to estimate the area under a curve by applying straight line segments. It's easy to use but might not be as accurate with a small number of segments.
- **Simpson's Rule**: Employs parabolic arcs instead of straight lines, providing more precision especially for functions that are smooth or nearly quadratic.

### Choosing the Right Method
- If the function is highly linear or if you require a quick solution with reasonable accuracy, the Trapezoidal Rule may suffice.
- For functions that change rapidly or when higher accuracy is needed, Simpson’s Rule might be preferred due to its ability to fit the curve more closely.

These methods are foundational to numerical analysis and engineering, often implemented in computational software to assist in scenarios where analytical solutions are impractical. The choice between different approximation methods often balances between computational efficiency and the required precision level of the approximation. By understanding these nuances, choosing the correct method for any given problem becomes an informed decision.