Problem 62 - Calculus Single Variable

Question

A function \(f\) and an interval \([a, b]\) are specified. Calculate the Simpson's Rule approximations of \(\int_{a}^{b} f(x) d x\) with \(N=10\) and \(N=20 .\) If the first five decimal places do not agree, increment \(N\) by \(10 .\) Continue until the first five decimal places of two consecutive approximations are the same. State your answer rounded to four decimal places. $$ f(x)=\exp (\sqrt{x}) $$

Step-by-Step Solution

Answer

Use Simpson's Rule to calculate integrals iteratively with increasing intervals until results stabilize; round the final result to four decimal places.

Step 1: Understanding Simpson's Rule

Simpson's Rule is a numerical method to approximate the definite integral of a function. It provides a more accurate approximation than the trapezoidal rule by considering parabolic segments. The formula is given by: \[S_N = \frac{h}{3}\left[f(x_0) + 4\sum_{i=1,3,5,...}^{N-1} f(x_i) + 2\sum_{i=2,4,6,...}^{N-2} f(x_i) + f(x_N)\right],\]where \(h = \frac{b-a}{N}\), and \(x_i = a + ih\). We need to ensure increasing \(N\) until the results stabilize within five decimal places.

Step 2: Initial Setup for N=10

First, calculate with \(N=10\):1. Compute \(h: h = \frac{b-a}{10}\).2. Determine the partition points: \(x_i = a + i \cdot h\) for \(i = 0, 1, 2, ..., 10\).3. Evaluate \(f(x_i) = e^{\sqrt{x_i}}\).4. Apply the Simpson's rule formula to find \(S_{10}\).

Step 3: Calculation for N=10

Carry out calculations for \(N=10\):1. Find \(h\), then use the function \(f(x) = e^{\sqrt{x}}\) to find values at each \(x_i\).2. Sum values according to Simpson's Rule: - Sum of all \(f(x_i)\) for odd \(i\) (multiply by 4). - Sum of all \(f(x_i)\) for even \(i\) (multiply by 2). - Add \(f(x_0)\) and \(f(x_{10})\).3. Use these in Simpson's formula to compute \(S_{10}\).

Step 4: Increase to N=20

Repeat the above steps with \(N=20\):1. Recalculate \(h = \frac{b-a}{20}\).2. Re-evaluate function values at new \(x_i\) points.3. Apply the Simpson's Rule again to find \(S_{20}\).

Step 5: Compare the Results

Compare the results of \(S_{10}\) and \(S_{20}\). If the first five decimals do not match, increase \(N\) by 10 and recalculate. This process ensures convergence to a more accurate approximation.

Step 6: Increment N and Repeat

Keep increasing \(N\) by 10, calculating the new approximation each time until the first five decimal places of successive results match. Calculate \(S_{30}\), \(S_{40}\), etc., as needed, until the precision is stable.

Step 7: Finalize and Round to Four Decimals

Once the results stabilize to five decimal places, take the last approximation, and round it to four decimals to present the final result.

Key Concepts

Numerical Integration

Numerical integration is an essential tool in calculus used to estimate the value of a definite integral when an exact solution is not readily obtainable. This approach is particularly valuable when dealing with functions that are difficult to integrate analytically.
Techniques like Simpson's Rule fall under numerical integration. They help in approximating the area under a curve by summing up segments of simpler geometric shapes. Numerical integration provides a means to analyze real-world situations where precise calculations are required, yet difficult to achieve with standard methods.
Some of the most commonly used numerical integration methods include:

• Trapezoidal Rule
• Simpson's Rule
• Midpoint Rule

Simpson's Rule offers improved accuracy by using parabolic segments instead of linear ones to model the curve within a given interval.

Definite Integral Approximation

Definite integral approximation involves estimating the integral of a function over a specific interval, which is critical in applications where precision is vital.
In the context of Simpson's Rule, we approximate the definite integral by dividing the interval into smaller sub-intervals, and then summing up areas under quadratic pieces, effectively allowing us to calculate with ease.
The approximation can be fine-tuned by adjusting the number of sub-intervals. In the exercise, the intervals were increased incrementally from 10 to 20, and further as needed, to achieve a more precise approximation.
This step-wise approach of incrementing sub-intervals helps minimize approximation error and converge toward the exact integral value, ensuring that calculations align closely with mathematical reality.

Convergence Analysis

Convergence analysis in numerical integration is vital for guaranteeing that our approximations approach the real integral value as we refine our calculations. This step ensures that the integration method's result becomes stable and accurate with increased segments or sub-intervals.
In our exercise, we monitored the results of successive approximations, using higher values for sub-intervals (N=10, N=20, etc.). The precise aim was to stabilize the first five decimal places of two consecutive results.
This stabilizing behavior indicates convergence. As we observe the outcomes from methods like Simpson's Rule, their effectiveness can be trusted as they approximate the real integral value more accurately over repeated iterations.

Error Estimation

Error estimation is an integral part of any numerical approximation, as it provides insight into the potential inaccuracies in the computed results. This concept helps users understand how close their approximations are to the true values and decide effectively how many sub-intervals are necessary for reliable results.
For Simpson’s Rule, the error in approximation drops significantly as the number of intervals increases due to its quadratic convergence feature. The process involves checking the early calculated results against more refined computations to observe the decrease in error.
This exercise of monitoring differences between successive approximations, like with incremental N values, helps pinpoint where significant improvements occur. This understanding guides in adjusting our computational efforts to achieve a balance between computational cost and desired accuracy.