Problem 64 - 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. $$ f(x)=\sin (\pi \cos (x)) \quad[0, \pi / 3] $$

Step-by-Step Solution

Answer

The integral is approximately equal to 0.5989 evaluated to four decimal places for sufficiently large N such that the first five decimal places agree.

Step 1: Understand Simpson's Rule

Simpson's Rule is a method for numerical integration that approximates the integral of a function using quadratic polynomials. For a given interval \([a, b]\) divided into \(N\) subintervals, where \(N\) is even, the rule is given by \[ S_N = \frac{b-a}{3N} \left( f(x_0) + 4 \sum_{i\text{ odd}} f(x_i) + 2 \sum_{i\text{ even}} f(x_i) + f(x_N) \right) \] where \(x_i = a + i\frac{b-a}{N}\).

Step 2: Set Up the Interval and Function

Given the function \(f(x) = \sin(\pi \cos(x))\) on the interval \([0, \pi/3]\), we need to calculate the integral approximation using Simpson's Rule for \(N = 10\) and then for other values incremented as needed.

Step 3: Calculate Simpson's Rule for N=10

Divide the interval \([0, \pi/3]\) into 10 subintervals: \(x_0, x_1, ..., x_{10}\). Calculate each \(x_i\) as \(x_i = \frac{i \pi}{30}\). Evaluate \(f(x_i) = \sin (\pi \cos(x_i))\) for each \(i\). Apply Simpson's Rule formula to compute \(S_{10}\).

Step 4: Calculate Simpson's Rule for N=20

Repeat the process for \(N=20\), dividing the interval into 20 subintervals: \(x_0, x_1, ..., x_{20}\). Calculate \(x_i = \frac{i \pi}{60}\) and evaluate \(f(x_i) = \sin (\pi \cos(x_i))\) for each \(i\). Use the formula to find \(S_{20}\).

Step 5: Compare and Adjust N

After computing \(S_{10}\) and \(S_{20}\), compare the results rounded to five decimal places. If they do not match, increment \(N\) by 10 and repeat the calculations until the first five decimal places of two consecutive approximations agree.

Step 6: Finalize the Answer

Continue calculating Simpson's Rule for increasing \(N\) until \(N = 30\) and then \(N = 40\). Once two consecutive results match to five decimal places, finalize your answer by rounding it to four decimal places.

Key Concepts

Numerical Integration

Numerical integration is a powerful technique used to find approximate values of definite integrals. When the actual integral is complex or impossible to solve analytically, we turn to numerical methods. Simpson's Rule is one such numerical technique.
It provides a way to approximate the area under a curve by using quadratic polynomials. This rule divides the interval of integration into an even number of segments. Then, it fits a parabola to every two segments to estimate the area more accurately.
Here are some key points:

• It is especially useful for smooth functions.
• Simpson’s Rule tends to be more accurate than other numerical methods like the Trapezoidal Rule.
• It requires an even number of subintervals for the approximation.

Simpson's Rule is most effective when the function behaves nicely over the interval. It’s a fabulous method when it’s tedious or impossible to determine the integral using elementary functions.

Approximation of Integrals

Approximating integrals means finding the approximate area under a curve. This approximation is significant in many real-world applications, from physics to economics.
In this exercise, the function given is rather intricate, making direct integration challenging. Thus, Simpson’s Rule serves as a practical method to achieve an approximation.
To understand why approximation is pivotal:

• Not all functions have elementary antiderivatives. Approximations are necessary for these cases.
• A good approximation like Simpson's Rule, often achieves close results with relatively few intervals.
• For theoretical computations, exact results might not be practical or needed.

By employing Simpson's Rule, you gain not only a numerical value but also an appreciation for how integral calculus can be harnessed to solve complex problems effectively. The task specifies adjusting the number of subintervals until the approximation stabilizes, ensuring higher precision.

Quadratic Polynomials in Calculus

Quadratic polynomials play a crucial role in calculus, especially in numerical methods like Simpson’s Rule. The idea here is to approximate curved segments of the function with parabolic arcs.
Parabolas, being quadratic in nature, are versatile and provide a nearer fit than simple linear segments, especially when it comes to smooth and continuous functions.
Why use parabolas?

• They can capture the curvature of functions better than straight lines.
• Their structure allows for easier integration over subintervals.
• They are fundamental in understanding how shapes can approximate more complex curves.

Simpson’s Rule benefits the most from quadratic approximation, as it effectively leverages the parabola’s properties to mimic the function’s path between points. The mathematics underlying this involves understanding how quadratic expressions can reflect a function’s gradient and curvature over small segments.