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Chapter 7: Problem 15

Use Table 7.9 $$\begin{array}{c|c|c|c|c|c} \hline t & 0.0 & 0.1 & 0.2 & 0.3 & 0.4 \\ \hline g(t) & 1.87 & 2.64 & 3.34 & 3.98 & 4.55 \\ \hline t & 0.5 & 0.6 & 0.7 & 0.8 & 0.9 \\ \hline g(t) & 5.07 & 5.54 & 5.96 & 6.35 & 6.69 \\\ \hline \end{array}$$ Estimate \(\int_{0}^{0.9} g(t) d t\) using a right-hand sum with \(n=3.\)

Short Answer

Expert verified
\(\int_{0}^{0.9} g(t) dt \approx 4.863\) with right-hand sum and \(n=3\).

Step by step solution

01

Understanding the Right-Hand Sum

A right-hand sum estimates the integral of a function by dividing the interval into equal segments, using the function's value at the right endpoint of each segment. With \(n=3\), you'll break the interval \([0, 0.9]\) into 3 equal sub-intervals: \([0, 0.3], [0.3, 0.6], [0.6, 0.9]\).
02

Determine \\ \(\Delta t\\ \)

Calculate the width of each sub-interval, \(\Delta t\). Since \([0, 0.9]\) is divided into 3 intervals, \(\Delta t = \frac{0.9 - 0}{3} = 0.3\).
03

Identify Right Endpoints

Identify the right endpoints for each sub-interval: 0.3, 0.6, and 0.9. At these points, you'll evaluate \(g(t)\) using Table 7.9.
04

Evaluate \\ \(g(t)\\ \) at Right Endpoints

Using Table 7.9, find the following values: \(g(0.3) = 3.98\), \(g(0.6) = 5.54\), and \(g(0.9) = 6.69\).
05

Calculate the Right-Hand Sum

The right-hand sum is calculated as the sum of the products of each function value and \(\Delta t\):\[ R = \Delta t \times (g(0.3) + g(0.6) + g(0.9)) = 0.3 \times (3.98 + 5.54 + 6.69) \].
06

Solve the Equation

Calculate the sum of the function values and then multiply by \(\Delta t\):\[ R = 0.3 \times (3.98 + 5.54 + 6.69) = 0.3 \times 16.21 = 4.863 \].

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Key Concepts

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

Numerical Integration
Numerical integration is a powerful method to approximate the integral of a function, particularly when finding the exact integral is challenging. Instead of using calculus to determine the area under a curve, numerical integration employs a discrete set of data points to make an approximation.
  • This approach is useful when dealing with complex functions or when only certain function values are known.
  • There are several techniques for numerical integration, such as the trapezoidal rule, Simpson's rule, and in our case, the right-hand sum.
The right-hand sum makes use of the values of the function at the right endpoints of sub-intervals to estimate the total area under the curve. This method is straightforward, requiring only basic arithmetic and a table of function values, making it accessible and easy for students to apply.
Sub-intervals
In numerical integration, dividing the interval over which you are integrating into smaller, manageable sub-intervals is key. This is because each sub-interval represents a small portion of the whole area under the curve.
  • For the right-hand sum method, these sub-intervals should have equal widths to simplify calculations.
  • The width of each sub-interval, denoted as \( \Delta t \), is calculated by taking the difference between the start and end of the interval and dividing by the number of sub-intervals (\( n \)).
In our example, with the interval \[ [0, 0.9] \] and \( n = 3 \), each sub-interval has a width of 0.3. By organizing the interval into parts \[ [0, 0.3], [0.3, 0.6], [0.6, 0.9] \], it becomes easier to apply the right-hand sum method efficiently.
Function Values
Function values are the core data in numerical integration methods like the right-hand sum. These values are obtained from the function being integrated over specific points.
  • In our approach, you focus on the right endpoints of each sub-interval.
  • The values of the function at these points dictate the 'height' of the rectangles used in the approximation.
For the example provided, we use the values from Table 7.9. Specifically, the function values at the right endpoints 0.3, 0.6, and 0.9 are utilized: \( g(0.3) = 3.98 \, g(0.6) = 5.54 \, g(0.9) = 6.69 \). These values are carefully chosen as they represent the best estimate of the function's behavior just before the next sub-interval starts.
Integral Estimation
Estimating an integral numerically, such as using the right-hand sum, involves calculating an approximate value representing the area under a function's curve. This is especially important when the function lacks a simple anti-derivative or when working with non-continuous data points.
  • The estimation requires you to sum up the products of the width of the sub-intervals and the function values at the right endpoints.
  • In our example, the equation arises as: \[ R = \Delta t \times (g(0.3) + g(0.6) + g(0.9)) = 0.3 \times (3.98 + 5.54 + 6.69) \]
The calculation continues with finding the sum of values, yielding 16.21. Multiplying by the width, 0.3, gives the final integral estimation: \( R = 4.863 \). This approach offers a practical solution for determining integral values in various scenarios, from engineering to finance.

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