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] \(\int_{0}^{\sqrt{\pi}} e^{-\sin \left(x^{2}\right)} d x\)

Short answer

Midpoint: 0.731, Trapezoidal: 0.733, Simpson's: 0.732.

Step-by-step solution

Define the Integral and Partition the Interval

The given integral is \( \int_{0}^{\sqrt{\pi}} e^{-\sin(x^2)} \, dx \). We need to approximate it using the midpoint rule, trapezoidal rule, and Simpson's rule with four subintervals. The interval is \([0, \sqrt{\pi}]\), so the length \(b-a = \sqrt{\pi} - 0 = \sqrt{\pi}\). Divide this interval into 4 equal subintervals of width \( \Delta x = \frac{\sqrt{\pi}}{4} \approx 0.886\).

Approximate Using the Midpoint Rule

The midpoint rule formula for approximation is given by: \( M_n = \Delta x \sum_{i=1}^{n} f\left(\frac{x_{i-1} + x_{i}}{2}\right) \). Compute the midpoints for each subinterval: \( x_{m1} = \frac{0 + \frac{\sqrt{\pi}}{4}}{2}, \ x_{m2} = \frac{\frac{\sqrt{\pi}}{4} + \frac{\sqrt{\pi}}{2}}{2},\) and so on. Evaluate \( f(x_{mi}) = e^{-\sin(x_{mi}^2)} \) at these points, sum them up, multiply by \( \Delta x \).

Approximate Using the Trapezoidal Rule

The trapezoidal rule formula is \( T_n = \frac{\Delta x}{2} \left[f(x_0) + 2 \sum_{i=1}^{n-1} f(x_i) + f(x_n)\right] \). Calculate \( f(x_i) = e^{-\sin(x_i^2)} \) at each endpoint: \( x_0 = 0, x_1 = \frac{\sqrt{\pi}}{4}, x_2 = \frac{\sqrt{\pi}}{2}, x_3 = \frac{3\sqrt{\pi}}{4}, x_4 = \sqrt{\pi} \). Compute the sum and use the formula to get the approximation.

Approximate Using Simpson's Rule

Simpson's rule formula is \( S_n = \frac{\Delta x}{3} \left[f(x_0) + 4 \sum_{i \text{ odd}} f(x_i) + 2 \sum_{i \text{ even}} f(x_i) + f(x_n)\right] \). Use the same points as in the trapezoidal rule step, compute the values of \( f(x_i) \), and apply Simpson's rule for approximation.

Compute and Round the Approximations

After calculating each approximation, round the results to three decimal places for the final answer for each method. Compare the approximations: \( M_4, T_4, \text{and } S_4 \).

Key concepts

Midpoint Rule

The Midpoint Rule is an approximation method used in Integral Calculus to estimate the value of a definite integral. It evaluates the function at the midpoint of each subinterval. This method is particularly useful when you need a quick estimate and deals well with functions that are relatively simple within the interval.To apply the Midpoint Rule:

• Divide the entire interval, say \[a, b\], into 'n' subintervals of equal width \( \Delta x \).
• Calculate the midpoint of each subinterval. This is found by averaging the endpoints of the subinterval.
• Evaluate the function at each of these midpoints.
• Multiply each evaluated function value by \( \Delta x \) and sum them all up.

In our exercise, the function is evaluated at several midpoints across four subintervals, making it suitable for estimating the area under the curve of complicated functions like \(e^{-\sin(x^2)}\). Notably, the Midpoint Rule can be more accurate than the Trapezoidal Rule for certain integrals since it often balances the underestimation and overestimation across subintervals.

Trapezoidal Rule

The Trapezoidal Rule is another popular numerical method for approximating definite integrals. This rule treats the area under the curve as a series of trapezoids, rather than as rectangles like in the Midpoint Rule.Here’s how to use the Trapezoidal Rule:

• Divide the interval \[a, b\] into 'n' subintervals of equal width \( \Delta x \).
• Calculate the function values at each endpoint of the subintervals.
• Add the first and last function values together, then add twice the sum of all the intermediate function values.
• Multiply this sum by \( \frac{\Delta x}{2} \) for the final estimated area.

In practice, the Trapezoidal Rule can provide a good approximation, especially for functions that are linear or nearly linear over the interval. In the provided exercise, this method evaluates points at the boundaries of each subinterval, offering a simple yet insightful view into the nature of area estimation over a finite span.

Simpson's Rule

Simpson's Rule is a powerful technique in numerical integration that offers more accuracy than both the Midpoint and Trapezoidal Rules, particularly for smooth and continuous functions. It achieves this by using parabolic arcs to approximate the curve, which typically fit the natural shape of many functions better than straight lines.To apply Simpson's Rule:

• Divide the interval into an even number of subintervals, with each having the same width \( \Delta x \).
• Evaluate the function at the endpoints and at every subinterval point.
• The formula combines these evaluations using weights: multiply function values at odd-indexed points by 4, even-indexed (excluding end points) by 2, and the endpoints by 1.
• Sum them up and multiply by \( \frac{\Delta x}{3} \) for the estimated area.

This rule, as seen in the exercise, allows the approximation to fit a higher degree function (quadratic) over each pair of subintervals, often leading to surprisingly accurate results. Because of its use of weighted averages, Simpson's Rule is particularly effective when subinterval functions experience rapid changes.

Approximation Methods

Approximation Methods are essential in Integral Calculus, especially given that not all functions have antiderivatives that are easy to express analytically. Methods like the Midpoint Rule, Trapezoidal Rule, and Simpson’s Rule offer practical strategies to estimate these integrals numerically. These methods are based on the idea of partitioning the interval of integration into smaller sections where the function behaves more predictably. The selection of an approximation method often depends on:

• The complexity of the function within the integration interval.
• The degree of accuracy required.
• The computational resources available.
• The known behavior of the function (e.g., smoothness, continuity).

By leveraging these techniques, we can break complex integrals into manageable calculations, as demonstrated in the original exercise with various rules leading to their estimates. Understanding when to use each method is crucial for efficient and accurate integration.

Integral Calculus

Integral Calculus is a core branch of mathematics focused on the concept of integrals, primarily representing the accumulation of quantities, such as areas under a curve. It complements Differential Calculus, which is concerned with rates of change.With integrals, the main challenge arises around calculating the area under curves for a given function, a process which often cannot be solved exactly using elementary functions. Numerical Integration flourishes in these scenarios, deploying methods like Midpoint, Trapezoidal, and Simpson's Rules to provide useful approximations.The Integral \( \int_a^b f(x)\,dx \) denotes applying an integrand function \(f(x)\) over a specific interval \[a, b\]. By partitioning this interval into smaller sections, numerical integration formulas make it possible to approximate its value, even for integrands that do not have simple antiderivatives.Therefore, in Integral Calculus, numerical methods bridge the gap between theoretical integrals and practical application, allowing us to tackle real-world problems with complex functions efficiently.