Problem 48 - Calculus Single Variable

Question

Calculate \(\mathcal{R}\left(f, \mathcal{L}_{N}\right),\) the Riemann sum \(\mathcal{R}(f, \mathcal{S})\) using the midpoint of each subinterval for the choice of points, and \(\mathcal{R}\left(f, \mathcal{U}_{N}\right)\). (You will notice that the inequalities of line \((5.2 .2)\) hold. \()\) $$ f(x)=\sin (x) \quad I=[0, \pi / 2], N=2 $$

Step-by-Step Solution

Answer

The midpoint Riemann sum is approximately 1.026, and the upper Riemann sum is approximately 1.34.

Step 1: Define the Interval and Number of Subintervals

The interval given is \( I = [0, \frac{\pi}{2}] \), and \( N = 2 \) represents the number of subintervals. Since the interval is divided into 2 subintervals, each subinterval will have a width \( \Delta x \) given by \( \Delta x = \frac{\text{Length of } I}{N} = \frac{\pi}{2N} = \frac{\pi}{4} \).

Step 2: Determine Midpoints of Subintervals

For midpoint Riemann sums, choose points \( x_i^* \) which are the midpoints of each subinterval. The subintervals are \([0, \frac{\pi}{4}]\) and \([\frac{\pi}{4}, \frac{\pi}{2}]\). The midpoints are: \( x_1^* = \frac{\pi}{8} \) and \( x_2^* = \frac{3\pi}{8} \).

Step 3: Evaluate the Function at Midpoints

Calculate the function value at each midpoint: \( f(x_1^*) = \sin(\frac{\pi}{8}) \) and \( f(x_2^*) = \sin(\frac{3\pi}{8}) \). We use trigonometric identities or a calculator to find these values approximately: \( \sin(\frac{\pi}{8}) \approx 0.383 \) and \( \sin(\frac{3\pi}{8}) \approx 0.924 \).

Step 4: Compute the Midpoint Riemann Sum

The midpoint Riemann sum is calculated as: \[ \mathcal{R}(f, \mathcal{S}) = \sum_{i=1}^{N} f(x_i^*) \Delta x \] \[ \mathcal{R}(f, \mathcal{S}) = \left(\sin\left(\frac{\pi}{8}\right) + \sin\left(\frac{3\pi}{8}\right)\right) \frac{\pi}{4} \approx (0.383 + 0.924) \times \frac{\pi}{4} \approx 1.307 \times 0.785 = 1.026 \]

Step 5: Determine Upper Sum Partition Points

For the upper sum (using right endpoints), choose points at the right endpoint of each subinterval. The points are \( x_2 = \frac{\pi}{4} \) and \( x_3 = \frac{\pi}{2} \).

Step 6: Evaluate the Function at Right Endpoints

Calculate the function value at each right endpoint: \( f(x_2) = \sin(\frac{\pi}{4}) \) and \( f(x_3) = \sin(\frac{\pi}{2}) \). These give values \( \sin(\frac{\pi}{4}) = \frac{\sqrt{2}}{2} \approx 0.707 \) and \( \sin(\frac{\pi}{2}) = 1 \).

Step 7: Compute the Upper Riemann Sum

The upper Riemann sum is: \[ \mathcal{R}(f, \mathcal{U}_N) = \sum_{i=1}^{N} f(x_{i+1}) \Delta x \] \[ \mathcal{R}(f, \mathcal{U}_N) = \left(\sin\left(\frac{\pi}{4}\right) + \sin\left(\frac{\pi}{2}\right)\right) \frac{\pi}{4} \approx (0.707 + 1) \times 0.785 \approx 1.707 \times 0.785 \approx 1.34 \]

Key Concepts

Midpoint Rule

The Midpoint Rule is a method used in numerical integration to approximate the definite integral of a function. This approach divides the integration interval into several subintervals. For each subinterval, instead of using the endpoints, the value of the function is evaluated at the midpoint.
This technique is especially useful because it often provides a better approximation of the integral compared to using endpoints. Here's how it works in practice:

• First, the interval is divided into smaller subintervals of equal width.
• The midpoints of these subintervals are calculated, and the function value at each midpoint is determined.
• These function values are multiplied by the width of the subintervals to approximate the area under the curve.

When using the Midpoint Rule, it is beneficial because it can counteract some errors that might arise when using other Riemann sum approaches. This makes it a preferred choice for obtaining an integral estimate when the function exhibits certain symmetrical properties.

Trigonometric Functions

Trigonometric functions are fundamental in mathematics, particularly when dealing with angles and periodic phenomena. The three primary trigonometric functions are sine, cosine, and tangent, each relating to angles in a right triangle and their corresponding ratios of side lengths.
In the context of numerical integration, specifically with Riemann sums, evaluating trigonometric functions at specific points is crucial. Consider the function given in the exercise, \( f(x) = \sin(x) \). The sine function oscillates between -1 and 1, making it periodic. Because of this property, calculating integrals for portions of a sine wave involves:

• Understanding sine values at critical points (like \( \frac{\pi}{4} \) or \( \frac{\pi}{2} \)).
• Utilizing identities—such as \( \sin(\frac{\pi}{2}) = 1 \) and \( \sin(0) = 0 \)—to simplify computations.
• Approximating these values using technology when needed.

By grasping both the properties and evaluations of trigonometric functions, students can efficiently carry out numerical integration through methods like the Midpoint Rule.

Numerical Integration

Numerical integration is a technique used to approximate the value of definite integrals. When an integral cannot be solved analytically, numerical methods come into play. These methods approximate the area under a curve by dividing the interval into segments, or subintervals.
Some common numerical integration methods include:

• Riemann Sums: This involves calculating the area under the curve by summing up areas of rectangles, using either the left, right, or midpoint values of each subinterval.
• Trapezoidal Rule: Averages the left and right endpoint values for each subinterval, treating the area as a series of trapezoids instead of rectangles.
• Simpson's Rule: Utilizes parabolic segments to achieve higher accuracy for smooth functions.

Each technique has its own strengths and is chosen based on the function’s characteristics and the desired accuracy. In essence, numerical integration allows us to handle complex integrals by breaking them down into manageable parts, providing reliable approximations where direct computation becomes impractical.