Let $f$ Be The Function Defined By $f(x)=2 \ln (x$\]. If Three Subintervals Of Equal Length Are Used, What Is The Value Of The Trapezoidal Sum Approximation For $\int_3^{4.5} 2 \ln (x) \, Dx$? Round To The Nearest Thousandth

Introduction

The trapezoidal rule is a numerical method used to approximate the value of a definite integral. It is based on the idea of approximating the area under a curve by dividing it into small trapezoids and summing the areas of these trapezoids. In this article, we will use the trapezoidal rule to approximate the value of the definite integral $\int_3^{4.5} 2 \ln (x) \, dx$.

The Function and the Integral

The function $f(x) = 2 \ln (x)$ is a natural logarithmic function with a base of $e$. The definite integral $\int_3^{4.5} 2 \ln (x) \, dx$ represents the area under the curve of this function between the limits $x = 3$ and $x = 4.5$.

The Trapezoidal Rule

The trapezoidal rule is based on the idea of approximating the area under a curve by dividing it into small trapezoids. The formula for the trapezoidal rule is:

$$\int_a^b f(x) \, dx \approx \frac{h}{2} \left[ f(a) + 2f(a+h) + 2f(a+2h) + \cdots + 2f(a+(n-1)h) + f(b) \right]$$

where $h$ is the width of each subinterval, $n$ is the number of subintervals, and $f(x)$ is the function being integrated.

Applying the Trapezoidal Rule

In this case, we are given that three subintervals of equal length are used to approximate the value of the definite integral. This means that $n = 3$ and $h = \frac{b-a}{n} = \frac{4.5-3}{3} = 0.5$.

We can now apply the trapezoidal rule to approximate the value of the definite integral:

$$\int_3^{4.5} 2 \ln (x) \, dx \approx \frac{0.5}{2} \left[ 2 \ln (3) + 2 \ln (3.5) + 2 \ln (4) + 2 \ln (4.5) \right]$$

Calculating the Trapezoidal Sum

To calculate the trapezoidal sum, we need to evaluate the function $f(x) = 2 \ln (x)$ at the points $x = 3, 3.5, 4, 4.5$.

$$f(3) = 2 \ln (3) \approx 2 \times 1.0986 \approx 2.1972$$

$$f(3.5) = 2 \ln (3.5) \approx 2 \times 1.2533 \approx 2.5066$$

$$f(4) = 2 \ln (4) \approx 2 \times 1.3863 \approx 2.7726$$

$$f(4.5) = 2 \ln (4.5) \approx 2 \times 1.5041 \approx 3.0082$$

We can now substitute these values into the formula for the trapezoidal sum:

$$\int_3^{4.5} 2 \ln (x) \, dx \approx \frac{0.5}{2} \left[ 2.1972 + 2.5066 + 2.7726 + 3.0082 \right]$$

$$\int_3^{4.5} 2 \ln (x) \, dx \approx \frac{0.5}{2} \left[ 10.4846 \right]$$

$$\int_3^{4.5} 2 \ln (x) \, dx \approx 2.5723$$

Conclusion

In this article, we used the trapezoidal rule to approximate the value of the definite integral $\int_3^{4.5} 2 \ln (x) \, dx$. We divided the interval into three subintervals of equal length and evaluated the function $f(x) = 2 \ln (x)$ at the points $x = 3, 3.5, 4, 4.5$. We then substituted these values into the formula for the trapezoidal sum and calculated the approximate value of the definite integral.

Rounding to the Nearest Thousandth

The approximate value of the definite integral is $2.5723$. Rounding this value to the nearest thousandth gives us $2.572$.

Final Answer

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Introduction

In our previous article, we used the trapezoidal rule to approximate the value of the definite integral $\int_3^{4.5} 2 \ln (x) \, dx$. In this article, we will answer some common questions related to the trapezoidal rule and its application to definite integrals.

Q: What is the trapezoidal rule?

A: The trapezoidal rule is a numerical method used to approximate the value of a definite integral. It is based on the idea of approximating the area under a curve by dividing it into small trapezoids and summing the areas of these trapezoids.

Q: How does the trapezoidal rule work?

A: The trapezoidal rule works by dividing the interval of integration into small subintervals of equal length. The function being integrated is then evaluated at the points of subdivision, and the areas of the trapezoids are calculated using the formula:

$$\int_a^b f(x) \, dx \approx \frac{h}{2} \left[ f(a) + 2f(a+h) + 2f(a+2h) + \cdots + 2f(a+(n-1)h) + f(b) \right]$$

where $h$ is the width of each subinterval, $n$ is the number of subintervals, and $f(x)$ is the function being integrated.

Q: What are the advantages of the trapezoidal rule?

A: The trapezoidal rule has several advantages, including:

• It is easy to implement and understand.
• It is relatively fast and efficient.
• It can be used to approximate the value of a definite integral to a high degree of accuracy.

Q: What are the disadvantages of the trapezoidal rule?

A: The trapezoidal rule has several disadvantages, including:

• It can be inaccurate for functions with sharp peaks or valleys.
• It can be sensitive to the choice of subinterval width.
• It can be computationally intensive for large intervals of integration.

Q: When should I use the trapezoidal rule?

A: You should use the trapezoidal rule when:

• You need to approximate the value of a definite integral to a high degree of accuracy.
• You have a function that is relatively smooth and does not have sharp peaks or valleys.
• You need to use a numerical method that is easy to implement and understand.

Q: Can I use the trapezoidal rule for functions with sharp peaks or valleys?

A: Yes, you can use the trapezoidal rule for functions with sharp peaks or valleys, but you may need to use a smaller subinterval width to achieve accurate results.

Q: How do I choose the subinterval width?

A: You should choose the subinterval width based on the smoothness of the function and the desired level of accuracy. A smaller subinterval width will generally result in a more accurate approximation, but it may also increase the computational time.

Q: Can I use the trapezoidal rule for functions with discontinuities?

A: Yes, you can use the trapezoidal rule for functions with discontinuities, but you may need to use a special technique to handle the discontinuity.

Conclusion

In this article, we answered some common questions related to the trapezoidal rule and its application to definite integrals. We discussed the advantages and disadvantages of the trapezoidal rule, and we provided guidance on when to use it and how to choose the subinterval width.

Final Answer

The final answer is $\boxed{2.572}$.