Problem 26 - Calculus Volume 1

Question

Express the following endpoint sums in sigma notation but do not evaluate them. $$R_{20} \text { for } f(x)=\sin x \text { on }[0, \pi]$$

Step-by-Step Solution

Answer

\(R_{20} = \sum_{i=1}^{20} \sin\left(i \cdot \frac{\pi}{20}\right) \cdot \frac{\pi}{20}\)

Step 1: Identify the Interval and Endpoint Values

We are given the interval \([0, \pi]\) and function \(f(x) = \sin x\). The goal is to express the endpoint sums \(R_{20}\) in sigma notation for these parameters.

Step 2: Determine the Subinterval Width

Since we need to divide the interval \([0, \pi]\) into 20 equal subintervals, the width of each subinterval, \(\Delta x\), is calculated by dividing the interval length by the number of subintervals: \(\Delta x = \frac{\pi - 0}{20} = \frac{\pi}{20}\).

Step 3: Establish the Right Endpoint Formula

For a right endpoint Riemann sum, each subinterval \(i\) (where \(i\) ranges from 1 to 20) has a right endpoint at \(x_i = 0 + i\Delta x = i \cdot \frac{\pi}{20}\).

Step 4: Write the Function in Terms of Right Endpoints

With the right endpoint formula, the function at each right endpoint is \(f(x_i) = \sin\left(i \cdot \frac{\pi}{20}\right)\).

Step 5: Construct the Sigma Notation

The Riemann sum \(R_{20}\) using sigma notation is expressed as:\[R_{20} = \sum_{i=1}^{20} \sin\left(i \cdot \frac{\pi}{20}\right) \cdot \Delta x\]This incorporates the function evaluated at each right endpoint and the width of each subinterval.

Key Concepts

Riemann Sum

A Riemann Sum is a method used in calculus to approximate the area under a curve, which can be thought of as the sum of areas of multiple rectangle-like slices beneath the curve. This approximation becomes more precise as the number of slices or rectangles increases. In our exercise, we are breaking down the interval \([0, \pi]\) for the function \(f(x) = \sin x\) into 20 smaller segments, thanks to the designation \(R_{20}\).

Important to remember:

• Riemann Sums can be categorized based on which point of the subinterval is used: left, right, or midpoint.
• The finer the division of segments within the interval, the closer the sum will be to the actual area under the curve.
• Each kind of Riemann Sum offers unique insights and approximations depending on your needs in analysis.

Right Endpoint

Choosing the right endpoint in the context of a Riemann Sum involves using the point at the far end of each subinterval to determine the height of each corresponding rectangle. When focusing on the right endpoint:

• The endpoint is calculated using the equation \(x_i = a + i \cdot \Delta x\), where \(a\) is the starting point of the interval and \(i\) varies from 1 to the total number of subintervals.
• This approach is particularly useful when you mainly have data at those specific endpoints, or for specific curves where their behavior at later points is significant.
• In our exercise, the right endpoint of each sub-segment results in using \(x_i = i \cdot \frac{\pi}{20}\) to find the values of \(\sin(x)\).

While being a straightforward approach, keep in mind that different endpoint choices (left, right, midpoint) can lead to varied approximations of the integral.

Subinterval Width

The subinterval width, denoted by \(\Delta x\), essentially refers to the width of each of the segments when dividing a given interval into smaller, equal parts.

For our tasks on the interval \([0, \pi]\):

• We need to divide the interval into 20 equal parts, which helps create the required sum to approximate the area under \(f(x) = \sin x\).
• Calculated by \(\Delta x = \frac{\text{interval length}}{\text{number of subintervals}}\), for this exercise, it becomes \(\frac{\pi}{20}\).
• Knowing this width is crucial, as it multiplies into each term within the sigma notation, scaling each rectangle area to approximate correctly.

An essential element in many calculus problems, the subinterval width ensures precision and control over the approximation process.

Trigonometric Function

Trigonometric functions, like \(\sin x\), play a fundamental role in mathematics, particularly when dealing with periodic phenomena and oscillations. In this exercise, we have:

• The function \(f(x) = \sin x\), which maps angles to the y-coordinate of a point on the unit circle.
• Its behavior is periodic, with a well-known wave-like shape that repeats every \(2\pi\).
• Within the Riemann Sum framework, evaluating at the right endpoint means taking precise values of \(\sin(x_i)\) which, in this context, becomes \(\sin\left(i \cdot \frac{\pi}{20}\right)\).
• These trig values directly incorporate into the sum within the sigma notation, influencing the final approximation result.

Trigonometric functions can often complicate calculus problems due to their periodic nature, but they also provide rich context and real-world applicability.