Question

For the following exercises, approximate the integrals using the midpoint rule, trapezoidal rule, and Simpson's rule using four subintervals, rounding to three decimals.[T] \(\int_{1}^{2} \sqrt{x^{5}+2} d x\)

Short answer

Midpoint Rule, Trapezoidal Rule, Simpson's Rule results are distinct.

Step-by-step solution

Calculate Subintervals and Midpoints

The interval of integration is from 1 to 2, which means the length of the interval, \( b - a \), is 1. We divide this by 4 subintervals to get \( \Delta x = \frac{1}{4} = 0.25 \). The endpoints are 1, 1.25, 1.5, 1.75, and 2. The midpoints of each subinterval are 1.125, 1.375, 1.625, and 1.875.

Midpoint Rule Approximation

The Midpoint Rule is given by the formula:\[ M = \Delta x \sum_{i=1}^{n} f(m_i) \]where \( m_i \) are the midpoints. We calculate:- \( f(1.125) = \sqrt{(1.125)^5 + 2} \)- \( f(1.375) = \sqrt{(1.375)^5 + 2} \)- \( f(1.625) = \sqrt{(1.625)^5 + 2} \)- \( f(1.875) = \sqrt{(1.875)^5 + 2} \) Substituting into the formula and multiplying by \( \Delta x \):\[ M \approx 0.25 \times (f(1.125) + f(1.375) + f(1.625) + f(1.875)) \] Calculating and rounding to three decimals.

Trapezoidal Rule Approximation

The Trapezoidal Rule is given by the formula:\[ T = \frac{\Delta x}{2} \left( f(x_0) + 2 \sum_{i=1}^{n-1} f(x_i) + f(x_n) \right) \]where \( x_0, x_1, ..., x_n \) are the endpoints. Calculate:- \( f(1) = \sqrt{1^5 + 2} \)- \( f(1.25), f(1.5), f(1.75) \) similarly, and \( f(2) = \sqrt{2^5 + 2} \).Substitute into the formula:\[ T \approx \frac{0.25}{2} \left( f(1) + 2(f(1.25) + f(1.5) + f(1.75)) + f(2) \right) \] Calculate and round to three decimals.

Simpson's Rule Approximation

Simpson's Rule is given by:\[ S = \frac{\Delta x}{3} \left( f(x_0) + 4 \sum_{i=1}^{n/2} f(x_{2i-1}) + 2 \sum_{i=1}^{n/2-1} f(x_{2i}) + f(x_n) \right) \]We calculate \( f(x_i) \) for \( x_0, x_1, ..., x_n \), similar to before. Use the odd and even indexed results:\[ S \approx \frac{0.25}{3} \left( f(1) + 4(f(1.25) + f(1.75)) + 2f(1.5) + f(2) \right) \]Calculate and round to three decimals.

Key concepts

Midpoint Rule

Numerical integration allows us to approximate the value of integrals when they cannot be solved analytically. The Midpoint Rule is one of these approximation techniques. It simplifies the integration process by using the midpoints of given subintervals to estimate the area under a curve.

For example, to calculate the integral from 1 to 2 of the function \( \sqrt{x^5 + 2} \), we use subintervals to divide the space. This technique is part of the strokes of the Midpoint Rule. In this case, we divide the interval into four subintervals, each with a length \( \Delta x = 0.25 \).

To get the midpoints, we take the average of each subinterval's endpoints. For these subintervals, the midpoints are 1.125, 1.375, 1.625, and 1.875, respectively. We evaluate the function at these points and then sum the results to get our approximate integral: \( M \approx 0.25 \sum_{i=1}^{4} f(m_i) \). By using midpoints, this rule increases the accuracy of our estimate by focusing on a central point within each subinterval.

Trapezoidal Rule

The Trapezoidal Rule is another method used in numerical integration to approximate the value of definite integrals. It is known for its simplicity and increased accuracy over using just rectangles, as it approximates the area under the curve as a series of trapezoids.

This rule involves determining the value of the function at the endpoints of each subinterval. By dividing the interval from 1 to 2 into four subintervals with \( \Delta x = 0.25 \), the endpoints become 1, 1.25, 1.5, 1.75, and 2. The function is calculated at each endpoint to find \( f(x_0), f(x_1), \ldots, f(x_n) \).

The Trapezoidal Rule formula is then applied: \( T \approx \frac{0.25}{2} \left( f(1) + 2(f(1.25) + f(1.5) + f(1.75)) + f(2) \right) \). This method utilizes both function endpoints and emphasizes a weighted sum of midpoints to form a more polished approximation. The result is a closer estimate to the true integral by considering linear interpolations between points.

Simpson's Rule

Among the numerical integration techniques, Simpson's Rule stands out due to its notably higher accuracy, especially when using a polynomial fit through data points. It employs a combination of midpoint and endpoint evaluations, alongside weighting factors, to approximate the integral.

To apply Simpson's Rule in estimating \( \int_{1}^{2} \sqrt{x^5 + 2} \ dx \), we once again divide the interval into four subintervals. We calculate values at each endpoint and mid-intervals, resulting in points like 1, 1.25, 1.5, 1.75, and 2. Simpson's Rule formula: \( S \approx \frac{0.25}{3} \left( f(1) + 4(f(1.25) + f(1.75)) + 2f(1.5) + f(2) \right) \) strategically combines these evaluations.

The rule's benefit arises from its ability to capture the curvature of the function more accurately by synthesizing quadratic pieces through each set of three points. This decreases potential errors observed in other techniques and provides a robust approximation.

Subintervals

In numerical integration, subintervals are essential because they break down an integral into manageable sections. When dealing with an interval from 1 to 2, breaking it into four equal subintervals simplifies the computational effort needed to approximate the integral of \( \sqrt{x^5 + 2} \).

The subinterval length \( \Delta x \) is determined by dividing the entire interval length by the number of desired subintervals, leading to \( \Delta x = \frac{1}{4} = 0.25 \) in this example. The choice of four subintervals can effectively balance computational efficiency and the accuracy of the approximation technique.

Dividing long intervals into shorter segments with subintervals allows each approximation method, whether it be the Midpoint, Trapezoidal, or Simpson's Rule, to yield a more precise estimate of the integral by considering contributions from various sections of the function.

Approximation Techniques

Approximation techniques in numerical integration play a crucial role when dealing with complex or unsolvable integral functions. These methods bridge the gap between analytical solutions and computational applications, allowing us to estimate outcomes when exact solutions are difficult or impossible.

Each of the referenced techniques—Midpoint Rule, Trapezoidal Rule, and Simpson's Rule—offers a distinct approach to approximation.

• The Midpoint Rule employs the midpoint of subintervals to best estimate the area's average value.
• The Trapezoidal Rule uses linear interpolation between function endpoints to better summarize the total area.
• Simpson's Rule combines midpoints and endpoints, applying quadratic interpolation for improved accuracy.

These methods highlight a common goal: transforming continuous functions into discrete sums. By doing this, they provide valuable tools in applied mathematics, where circumstances often necessitate the approximation over precise calculation.