Let $f$ Be The Function Defined By $f(x) = 3 \ln(x$\]. If Five Subintervals Of Equal Length Are Used, What Is The Value Of The Right Riemann Sum Approximation For $\int_2^3 3 \ln(x) \, Dx$? Round To The Nearest Thousandth If

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Introduction

The Riemann sum is a mathematical concept used to approximate the value of a definite integral. It is a method of approximating the area under a curve by dividing the area into small rectangles and summing the areas of the rectangles. In this article, we will discuss the right Riemann sum approximation for the definite integral $\int_2^3 3 \ln(x) \, dx$.

The Function $f(x) = 3 \ln(x)$

The function $f(x) = 3 \ln(x)$ is a logarithmic function with a base of $e$. The natural logarithm is a fundamental function in mathematics and is used to model many real-world phenomena. The function $f(x) = 3 \ln(x)$ is a simple transformation of the natural logarithm function, where the input is multiplied by $3$.

The Definite Integral

The definite integral $\int_2^3 3 \ln(x) \, dx$ represents the area under the curve of the function $f(x) = 3 \ln(x)$ between the limits of $x = 2$ and $x = 3$. The definite integral is a fundamental concept in calculus and is used to solve many problems in physics, engineering, and economics.

The Right Riemann Sum Approximation

The right Riemann sum approximation is a method of approximating the value of a definite integral by dividing the area under the curve into small rectangles and summing the areas of the rectangles. The right Riemann sum approximation is defined as:

$$\sum_{i=1}^{n} f(x_i) \Delta x$$

where $f(x_i)$ is the value of the function at the right endpoint of the $i$th subinterval, $\Delta x$ is the width of the subinterval, and $n$ is the number of subintervals.

Calculating the Right Riemann Sum Approximation

To calculate the right Riemann sum approximation for the definite integral $\int_2^3 3 \ln(x) \, dx$, we need to divide the area under the curve into five subintervals of equal length. The width of each subinterval is $\Delta x = \frac{3-2}{5} = \frac{1}{5}$.

The right endpoints of the subintervals are:

$x_1 = 2 + \frac{1}{5} = 2.2$ $x_2 = 2.2 + \frac{1}{5} = 2.4$ $x_3 = 2.4 + \frac{1}{5} = 2.6$ $x_4 = 2.6 + \frac{1}{5} = 2.8$ $x_5 = 2.8 + \frac{1}{5} = 3$

The values of the function at the right endpoints are:

$f(x_1) = 3 \ln(2.2) \approx 2.508$ $f(x_2) = 3 \ln(2.4) \approx 2.732$ $f(x_3) = 3 \ln(2.6) \approx 2.958$ $f(x_4) = 3 \ln(2.8) \approx 3.184$ $f(x_5) = 3 \ln(3) \approx 3.412$

The right Riemann sum approximation is:

$$\sum_{i=1}^{5} f(x_i) \Delta x \approx (2.508 + 2.732 + 2.958 + 3.184 + 3.412) \left(\frac{1}{5}\right) \approx 3.197$$

Conclusion

In this article, we discussed the right Riemann sum approximation for the definite integral $\int_2^3 3 \ln(x) \, dx$. We divided the area under the curve into five subintervals of equal length and calculated the value of the function at the right endpoints of each subinterval. The right Riemann sum approximation is approximately $3.197$, rounded to the nearest thousandth.

References

• [1] "Calculus" by Michael Spivak
• [2] "Introduction to Calculus" by Michael Sullivan
• [3] "Calculus: Early Transcendentals" by James Stewart
  

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Q&A: Right Riemann Sum Approximation

Q: What is the right Riemann sum approximation?

A: The right Riemann sum approximation is a method of approximating the value of a definite integral by dividing the area under the curve into small rectangles and summing the areas of the rectangles.

Q: How is the right Riemann sum approximation calculated?

A: The right Riemann sum approximation is calculated by dividing the area under the curve into subintervals of equal length, calculating the value of the function at the right endpoint of each subinterval, and summing the areas of the rectangles.

Q: What is the formula for the right Riemann sum approximation?

A: The formula for the right Riemann sum approximation is:

$$\sum_{i=1}^{n} f(x_i) \Delta x$$

where $f(x_i)$ is the value of the function at the right endpoint of the $i$th subinterval, $\Delta x$ is the width of the subinterval, and $n$ is the number of subintervals.

Q: How many subintervals are used in the right Riemann sum approximation?

A: In the example given in this article, five subintervals of equal length are used.

Q: What is the width of each subinterval?

A: The width of each subinterval is $\Delta x = \frac{3-2}{5} = \frac{1}{5}$.

Q: What is the value of the function at the right endpoints of the subintervals?

A: The values of the function at the right endpoints of the subintervals are:

$f(x_1) = 3 \ln(2.2) \approx 2.508$ $f(x_2) = 3 \ln(2.4) \approx 2.732$ $f(x_3) = 3 \ln(2.6) \approx 2.958$ $f(x_4) = 3 \ln(2.8) \approx 3.184$ $f(x_5) = 3 \ln(3) \approx 3.412$

Q: What is the right Riemann sum approximation for the definite integral $\int_2^3 3 \ln(x) \, dx$?

A: The right Riemann sum approximation for the definite integral $\int_2^3 3 \ln(x) \, dx$ is approximately $3.197$, rounded to the nearest thousandth.

Q: What is the purpose of the right Riemann sum approximation?

A: The purpose of the right Riemann sum approximation is to approximate the value of a definite integral by dividing the area under the curve into small rectangles and summing the areas of the rectangles.

Q: What are the advantages of the right Riemann sum approximation?

A: The advantages of the right Riemann sum approximation include:

• It is a simple and easy-to-understand method of approximating the value of a definite integral.
• It can be used to approximate the value of a definite integral for a wide range of functions.
• It can be used to approximate the value of a definite integral for a wide range of limits of integration.

Q: What are the disadvantages of the right Riemann sum approximation?

A: The disadvantages of the right Riemann sum approximation include:

• It is an approximation method, and the accuracy of the approximation depends on the number of subintervals used.
• It can be time-consuming to calculate the right Riemann sum approximation for a large number of subintervals.
• It may not be accurate for functions with a large number of oscillations or discontinuities.

Conclusion

In this article, we discussed the right Riemann sum approximation for the definite integral $\int_2^3 3 \ln(x) \, dx$. We answered a series of questions about the right Riemann sum approximation, including how it is calculated, what the formula is, and what the advantages and disadvantages are. We also provided a numerical example of the right Riemann sum approximation for the given definite integral.