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Magazine Chapter 5: Problem 26
Express the following endpoint sums in sigma notation but do not evaluate them. $$R_{20} \text { for } f(x)=\sin x \text { on }[0, \pi]$$
Short Answer
Expert verified
\(R_{20} = \sum_{i=1}^{20} \sin\left(i \cdot \frac{\pi}{20}\right) \cdot \frac{\pi}{20}\)
Step by step solution
01
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.
02
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: \(\Delta x = \frac{\pi - 0}{20} = \frac{\pi}{20}\).
03
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 \(x_i = 0 + i\Delta x = i \cdot \frac{\pi}{20}\).
04
Write the Function in Terms of Right Endpoints
With the right endpoint formula, the function at each right endpoint is \(f(x_i) = \sin\left(i \cdot \frac{\pi}{20}\right)\).
05
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.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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:
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 \(\sin(x)\).
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 \([0, \pi]\):
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.
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.
