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Chapter 1: Problem 30

In the following exercises, graph the function then use a calculator or a computer program to evaluate the following left and right endpoint sums. Is the area under the curve between the left and right endpoint sums? $$ \text { [T] } L_{50} \text { and } R_{50} \text { for } y=\frac{x+1}{x^{2}-1} \text { on the interval }[2,4] $$

Short Answer

Expert verified
Yes, the area is between \( L_{50} \) and \( R_{50} \).

Step by step solution

01

Understand the function and interval

The function we are working with is \( y = \frac{x+1}{x^2-1} \), and the interval of interest is \([2, 4]\). We need to find the area under this curve using the left and right endpoint sum approximations with 50 subintervals, denoted as \( L_{50} \) and \( R_{50} \).
02

Divide the interval into subintervals

To calculate \( L_{50} \) and \( R_{50} \), divide the interval \([2, 4]\) into 50 equal parts. Each subinterval has a width \(\Delta x = \frac{4-2}{50} = 0.04\).
03

Calculate \( L_{50} \)

The left approximation \( L_{50} \) is calculated by summing the areas of rectangles which use the left endpoint of each subinterval as the height. The sum can be expressed as \( L_{50} = \sum_{i=0}^{49} f(x_i) \cdot \Delta x \), where \( x_i = 2 + i \cdot 0.04 \). Use a calculator or program to compute \( L_{50} \).
04

Calculate \( R_{50} \)

The right approximation \( R_{50} \) is calculated by summing the areas of rectangles which use the right endpoint of each subinterval as the height. The sum is expressed as \( R_{50} = \sum_{i=1}^{50} f(x_i) \cdot \Delta x \), where \( x_i = 2 + i \cdot 0.04 \). Use a calculator or program to compute \( R_{50} \).
05

Analyze the area estimation

The true area under the curve is between the left endpoint sum \( L_{50} \) and the right endpoint sum \( R_{50} \). \( L_{50} \) is an underestimate and \( R_{50} \) is an overestimate, assuming the function is increasing on \([2, 4]\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Riemann Sums
Riemann sums are a fundamental concept in calculus used to estimate the area under a curve. This technique involves partitioning the area into a collection of rectangles and summing their areas to approximate the total area under the graph of a function. Here's the step-by-step insight into how Riemann sums work:
  • Partitioning the Interval: The interval \[a, b\] is divided into \ n \ smaller subintervals. The smaller the subintervals, the more accurate the approximation.
  • Choosing Sample Points: Within each subinterval, a sample point is chosen. This point will determine the height of the rectangle in that subinterval.
  • Calculating Rectangle Areas: Each rectangle's area is calculated by multiplying the width of the subinterval \(\Delta x\) by the height \(f(x_i)\).
  • Summing Rectangles: The sum of all rectangle areas gives the Riemann sum, which approximates the area under the curve.
Riemann sums can be further classified as left, right, or midpoint, depending on the sampling point used within the subintervals.
Left Endpoint Approximation
Left Endpoint Approximation is a specific type of Riemann sum that utilizes the left endpoint of each subinterval to determine the height of the rectangle. Here's how it works:
  • Left Endpoints: In the interval \[a, b\], for each subinterval, the left endpoint is taken as the point to evaluate the function. This value determines the height of the rectangle.
  • Height at Left Endpoint: For a function \(f\), the height of the rectangle for each subinterval is \(f(x_i)\), where \(x_i\) is the left endpoint.
  • Summing for Approximation: The sum of the areas of all these rectangles gives the left endpoint approximation \(L_n\).
This method generally gives an underestimate if the function is increasing, and an overestimate if the function is decreasing within the interval.
Right Endpoint Approximation
Right Endpoint Approximation is another type of Riemann sum that uses the right endpoint of each subinterval to compute the height of the rectangle. This method is similar to the left endpoint approximation but with its unique approach:
  • Right Endpoints: Instead of the left, the right endpoint of each subinterval in the interval \[a, b\] is used to determine the rectangle's height.
  • Height at Right Endpoint: Here, the height of each rectangle is given by \(f(x_{i+1})\), where \(x_{i+1}\) represents the right endpoint of the subinterval.
  • Summing for Approximation: By adding up the areas of all these rectangles, we get the right endpoint approximation \(R_n\).
If the function is increasing, the right endpoint sum typically results in an overestimate, while if the function is decreasing, it may yield an underestimate.
Numerical Integration
Numerical integration is a technique used to compute an approximation of a definite integral. It is especially useful when an exact analytical solution is challenging or impossible to obtain. It involves methods like Riemann sums and others for better accuracy. Numerical integration techniques include:
  • Trapezoidal Rule: This method improves upon Riemann sums by averaging the left and right endpoint approximations, forming a series of trapezoids instead of rectangles.
  • Simpson's Rule: This technique uses parabolic segments rather than straight lines to approximate the area under a curve, offering greater accuracy over both the trapezoidal rule and basic Riemann sums.
  • Monte Carlo Integration: This method employs random sampling to estimate the integral, which can be particularly useful for higher dimensions.
These approaches allow for precise area estimation under complex curves and are widely used in various scientific and engineering applications.

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Most popular questions from this chapter

In the following exercises, evaluate the definite integral. $$ \int_{\pi / 6}^{\pi / 2} \csc x d x $$

In the following exercises, use a suitable change of variables to determine the indefinite integral. $$ \int t\left(1-t^{2}\right)^{10} d t $$

[T] Compute the left and right Riemann sums, \(L_{10}\) and \(R_{10}\), and their average \(\frac{L_{10}+R_{10}}{2}\) for \(f(t)=\left(4-t^{2}\right)\) over \([1,2] .\) Given that \(\int_{1}^{2}\left(4-t^{2}\right) d t=1 . \overline{66},\) to how many decimal places is \(\frac{L_{10}+R_{10}}{2}\) accurate?

In the following exercises, \(f(x) \geq 0\) for \(a \leq x \leq b\). Find the area under the graph of \(f(x)\) between the given values \(a\) and \(b\) by integrating. $$ f(x)=2^{-x} ; a=3, b=4 $$

[T] The graph below plots the quadratic \(p(t)=6.48 t^{2}-80.31 t+585.69\) against the data in preceding table, normalized so that \(t=0\) corresponds to 1963. Estimate the average number of bald eagles per year present for the 37 years by computing the average value of \(p\) over [0,37]
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