Problem 43 - Calculus Single Variable

Question

Simpson's Rule is generally more accurate than the Midpoint Rule, but it is not always more accurate. Calculate \(A=\int_{-1}^{1} \sqrt{|x|} d x .\) With \(N=2,\) estimate \(A\) using both the Midpoint Rule, \(\mathcal{M}_{2},\) and Simpson's Rule, \(\mathcal{S}_{2} .\) What are the absolute errors? Repeat with \(N=4\).

Step-by-Step Solution

Answer

With N=2, the Midpoint Rule is more accurate. With N=4, Simpson's Rule is more accurate.

Step 1: Understanding the Problem

We need to estimate the integral \( A = \int_{-1}^{1} \sqrt{|x|} \, dx \) using both the Midpoint Rule and Simpson's Rule with \( N = 2 \) and \( N = 4 \). Additionally, we will calculate the absolute errors for each case.

Step 2: Exact Value of the Integral

Since we are estimating, let's compute the exact value of the integral: \( \int_{-1}^{1} \sqrt{|x|} \, dx \). As the function is even and the interval symmetric around 0, it simplifies to twice the integral from 0 to 1: \(2 \int_0^1 \sqrt{x} \, dx = 2 \left[ \frac{2}{3} x^{3/2} \right]_0^1 = \frac{4}{3}.\)

Step 3: Midpoint Rule with N=2

Divide the interval \([-1, 1]\) into two subintervals: \([-1, 0]\) and \([0, 1]\). The midpoint for \([-1, 0]\) is \(-0.5\) and for \([0, 1]\) is \(0.5\). The Midpoint Rule estimate is:\(\mathcal{M}_2 = \Delta x \sum_{i=1}^{2} f(x_i^*) = 1 \left( \sqrt{|-0.5|} + \sqrt{|0.5|} \right) = 1.414.\) The absolute error is \(|1.414 - \frac{4}{3}| = 0.0817.\)

Step 4: Simpson's Rule with N=2

For \( N = 2 \), the interval is \([-1, 1]\) divided at \( x=0 \). Simpson's Rule estimate is:\[\mathcal{S}_2 = \frac{1-(-1)}{6} \left( f(-1) + 4f(0) + f(1) \right) = \frac{1}{3} (1 + 0 + 1) = \frac{2}{3}.\] The absolute error is \(|\frac{2}{3} - \frac{4}{3}| = \frac{2}{3} = 0.6667.\) This shows it's not more accurate here.

Step 5: Midpoint Rule with N=4

Divide the interval \([-1, 1]\) into four subintervals: \([-1, -0.5], [-0.5, 0], [0, 0.5], [0.5, 1]\). The midpoints are \(-0.75, -0.25, 0.25, 0.75\). The Midpoint Rule estimate is:\(\mathcal{M}_4 = \frac{1}{2} \left( \sqrt{0.75} + \sqrt{0.25} + \sqrt{0.25} + \sqrt{0.75} \right) = 1.346.\) The absolute error is \(|1.346 - \frac{4}{3}| = 0.0207.\)

Step 6: Simpson's Rule with N=4

Now, divide interval \([-1, 1]\) into segments \([-1,-0.5], [-0.5, 0], [0,0.5], [0.5, 1]\).Simpson's Rule requires adjusting formula: \[\mathcal{S}_4 = \frac{1}{12} \left( f(-1) + 4f(-0.5) + 2f(0) + 4f(0.5) + f(1) \right) = 1.333.\] The absolute error is \(|1.333 - \frac{4}{3}| = 0.0003.\)

Step 7: Conclusion

With \(N=2\), the Midpoint Rule had a smaller error (0.0817) compared to Simpson's Rule (0.6667). With \(N=4\), Simpson's Rule produces a more accurate result (error 0.0003) than the Midpoint Rule (error 0.0207).

Key Concepts

Simpson's Rule

Simpson's Rule is a popular method in numerical integration used to estimate the value of definite integrals. It works by approximating the integrand with a piecewise quadratic function. This rule is particularly effective for smooth functions over small intervals.
Simpson's Rule formula can be expressed mathematically as:

• For two intervals: \[\mathcal{S}_2 = \frac{b-a}{6} \left[ f(a) + 4f\left(\frac{a+b}{2}\right) + f(b) \right]\]


• For four intervals:\[\mathcal{S}_4 = \frac{b-a}{12} \left[ f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + f(x_4) \right]\]

Here, \(x_0, x_1, \ldots, x_4\) are the evaluated points across the subintervals.
This method often produces more accurate results than the Midpoint Rule, especially with more subintervals, but as the exercise shows, it's not always superior when lesser subintervals are used.

Midpoint Rule

The Midpoint Rule is another widely used technique for approximating the integral of a function. It uses the midpoint of each subinterval to estimate the area under the curve. The simplicity of this rule makes it valuable for quick calculations.
It is expressed as:

• \[\mathcal{M}_N = \Delta x \sum_{i=1}^{N} f(x_i^*)\]

Where \(\Delta x\) is the width of the subintervals and \(x_i^*\) are the midpoints.
Although straightforward, this rule might not provide highly accurate results for functions with significant curvature. However, as seen in the case of \(N=2\), it outperformed Simpson’s Rule when applied to the integral \(\int_{-1}^{1} \sqrt{|x|} \, dx\).

Absolute Error

Absolute error allows us to measure how far our numerical estimate is from the exact value. It is calculated as the absolute value of the difference between the estimated integral and the exact integral.
Expressed as:

• \[| \text{estimated value} - \text{exact value} |\]

In the context of the exercise, the absolute error assists in identifying which method (Simpson's Rule or Midpoint Rule) provides a closer approximation for the given integral.
Although Simpson's Rule usually decreases error with more subintervals, it's interesting to note that for \(N=2\), it had a larger error compared to the Midpoint Rule. With \(N=4\), however, Simpson's Rule offered nearly exact precision.

Integral Estimation

Estimating integrals numerically is crucial when an integral doesn’t have a simple antiderivative or the function involves complexity. Numerical methods, such as Simpson's Rule and the Midpoint Rule, are tools used to achieve this.
Integral estimation is essential in various fields ranging from physics to engineering, where exact solutions are often complex or impossible.
Choosing the correct method for integral estimation depends on:

• The number of subintervals you plan to use.
• The nature and behavior of the function you are integrating.

In the case of symmetric intervals and functions, like the one used in our problem (\(\sqrt{|x|}\)), and depending on computation feasibility, one might choose to use a higher number of intervals to achieve a precise estimation using Simpson's Rule, despite its initial disadvantage at smaller intervals.