Problem 9 - Calculus

Question

In Problems \(7-10,\) use the methods of (1) left Riemann sum, (2) right Riemann sum, (3) midpoint Riemann sum, (4) Trapezoidal Rule, (5) Parabolic Rule with \(n=4,8,16\). (Note that none of these can be evaluated using the Second Fundamental Theorem of Calculus with the techniques you have learned so far.) Present your approximations in a table like this: $$ \int_{0}^{2} \sqrt{x^{2}+1} d x $$

Step-by-Step Solution

Answer

Compute the Riemann sums and trapezoidal and parabolic rules for each \(n\) set to 4, 8, and 16.

Step 1: Determine Subinterval Width

For an integral from 0 to 2 with different values of \(n\), the subinterval width \(\Delta x\) is calculated as \(\Delta x = \frac{b-a}{n}\). Here, \(a=0\) and \(b=2\). Hence, for different \(n\), \(\Delta x\) will be:- \(n=4: \Delta x = \frac{2-0}{4} = 0.5\)- \(n=8: \Delta x = \frac{2-0}{8} = 0.25\)- \(n=16: \Delta x = \frac{2-0}{16} = 0.125\)

Step 2: Left Riemann Sum

Calculate the left Riemann sum using \(n = 4, 8, 16\). The formula is:\[ L_n = \Delta x \sum_{i=0}^{n-1} f(x_i) \]Substitute the respective values of \(\Delta x\) and \(x_i\) to find \(L_4, L_8, L_{16}\). For instance, starting at \(x_0 = 0\), sum the function evaluations at the left endpoints of each subinterval.

Step 3: Right Riemann Sum

Calculate the right Riemann sum:\[ R_n = \Delta x \sum_{i=1}^{n} f(x_i) \]Using the same subinterval widths as in the left Riemann sum, substitute the values for \(x_i\) which are the right endpoints of each subinterval. Calculate for \(n = 4, 8, 16\) accordingly.

Step 4: Midpoint Riemann Sum

For the midpoint Riemann sum, the formula is:\[ M_n = \Delta x \sum_{i=0}^{n-1} f\left(\frac{x_i + x_{i+1}}{2}\right) \]This requires the value of the function at the midpoint of each subinterval. Compute for \(n = 4, 8, 16\).

Step 5: Trapezoidal Rule

The trapezoidal rule is given by:\[ T_n = \frac{\Delta x}{2} \left(f(x_0) + 2 \sum_{i=1}^{n-1} f(x_i) + f(x_n)\right) \]Calculate the approximation using this rule for \(n = 4, 8, 16\). Evaluate the function at each endpoint and weighted sum of the values in between.

Step 6: Parabolic Rule (Simpson's Rule)

Given by:\[ S_n = \frac{\Delta x}{3} \left(f(x_0) + 4 \sum_{i=1,3,5,...}^{n-1} f(x_i) + 2 \sum_{i=2,4,6,...}^{n-2} f(x_i) + f(x_n)\right) \]Only valid for even \(n\), so ensure you use this for \(n = 4, 8, 16\). Evaluate the sums where specified to find the approximation values.

Key Concepts

Riemann Sums

When we talk about numerical integration, Riemann Sums are a fundamental concept. They provide a way to approximate the value of an integral. Riemann Sums come in various types: left, right, and midpoint Riemann sums. Each has a slightly different approach to estimating the area under a curve.

Left Riemann Sum
- Uses the left endpoints of subintervals.
- If you have a partitioned interval \( [a, b] \) divided into \( n \) equal parts, the width of each subinterval is \( \Delta x = \frac{b - a}{n} \).
- You calculate the sum by evaluating the function at the left endpoint of each subinterval. Imagine stacking rectangles under the curve.

Right Riemann Sum
- Utilizes the right endpoints of subintervals.
- Like before, the interval is divided using \( \Delta x = \frac{b - a}{n} \).
- You get the sum by evaluating the function at the right endpoint, creating another arrangement of rectangles.

Midpoint Riemann Sum
- Looks at the middle point of each subinterval for function evaluation.
- This approach often provides more accuracy than the left or right methods.
- To calculate, find the average of each endpoint, evaluate the function, and sum these values. This can better approximate curves.

• Consider the midpoint sum to get a balanced approximation.
• All of these methods give a step toward finding the integral but may require many rectangles for precision.

Trapezoidal Rule

The Trapezoidal Rule is another way to approximate the value of a definite integral. Instead of using rectangles, this method fits trapezoids beneath the curve of the function.

- The interval \( [a, b] \) is divided into \( n \) equal parts with a subinterval width of \( \Delta x = \frac{b - a}{n} \).
- For each subinterval, a trapezoid is formed using the function values at the endpoints.
- The formula used is: \[ T_n = \frac{\Delta x}{2} \left(f(x_0) + 2 \sum_{i=1}^{n-1} f(x_i) + f(x_n)\right) \]

Why use trapezoids?

• Trapezoids fit naturally with linear sections of a curve, providing a better approximation when the graph is nearly straight.
• This method is often more accurate than Riemann sums when dealing with a small number of intervals.
• The Trapezoidal Rule can effectively approximate integrals for smooth and less erratic functions.

Simpson's Rule

When aiming for a high accuracy approximation, Simpson's Rule is a potent tool. It's often chosen for its increased precision, especially over an even number of subintervals. The technique resembles more of a parabolic approximation.

Here's how Simpson's Rule works:
- Divide the interval \( [a, b] \) into an even number of subintervals \( n \), with width \( \Delta x = \frac{b - a}{n} \).
- Instead of straight lines, approximations use parabolas to fit portions of the curve.
- The formula is:
\[ S_n = \frac{\Delta x}{3} \left(f(x_0) + 4 \sum_{i=1,3,5,...}^{n-1} f(x_i) + 2 \sum_{i=2,4,6,...}^{n-2} f(x_i) + f(x_n)\right) \]

Why prefer Simpson's Rule?

• The method leverages the power of quadratic functions to provide superior precision.
• Works best for functions that are smooth and "well-behaved" within the interval.
• Requires less \( n \) to achieve comparable accuracy to other methods.

Simpson’s Rule is a brilliant combination of algebra and analysis, illustrating how we can efficiently harness functional symmetry for integration.