Problem 32 - Calculus

Question

Use a graphing calculator to plot the graph of the integrand. Then estimate the integral as suggested in the margin note accompanying Theorem \(B\). $$ \int_{10}^{20}\left(1+\frac{1}{x}\right)^{5} d x $$

Step-by-Step Solution

Answer

The estimated integral is approximately 13.79 using the midpoint method.

Step 1: Understand the Function

We are given the integrand \( f(x) = \left(1+\frac{1}{x}\right)^{5} \). Understanding the function will help us predict the behavior of the graph and make an accurate estimation of the integral.

Step 2: Set Up Graphing Calculator

Enter the integrand \( f(x) = \left(1+\frac{1}{x}\right)^{5} \) into the graphing calculator and set the viewing window to encompass \( x = 10 \) to \( x = 20 \) for a clear visualization.

Step 3: Plot the Graph

Use the graphing calculator to plot the function over the interval \([10, 20]\). Observe how the function behaves on this interval to estimate the integral more accurately.

Step 4: Use Theorem B for Estimation

The margin note with Theorem B suggests we estimate the integral using the midpoint of the interval measuring its size. First, calculate the midpoint: \( x_{mid} = \frac{10 + 20}{2} = 15 \). Evaluate the function at this point: \( f(15) = \left(1 + \frac{1}{15}\right)^5 \).

Step 5: Calculate the Estimate

Compute \( f(15) \) which is approximately \( \left(1 + \frac{1}{15}\right)^5 = \left(\frac{16}{15}\right)^5 \approx 1.379 \). Multiply by the length of the interval (20 - 10 = 10): \( 1.379 \times 10 = 13.79 \).

Step 6: Final Integral Estimation

The estimated value for the integral \( \int_{10}^{20}\left(1+\frac{1}{x}\right)^{5} d x \) by using the midpoint method is approximately \( 13.79 \).

Key Concepts

Graphing Calculator

A graphing calculator is a vital tool in understanding and solving calculus problems, including integral estimation. It allows you to visualize complex functions and their behavior over specific intervals. By inputting the integrand, in this case, \( f(x) = \left(1+\frac{1}{x}\right)^5 \), into the calculator, you can plot its graph and see how it changes from one point to another.

When using a graphing calculator:

• Ensure you set the viewing window to cover the interval of interest; here, you need \( x = 10 \) to \( x = 20 \).
• This setup lets you focus on the relevant section of the graph for your integral estimation.

Visualizing the function helps you to better estimate the area under the curve, which represents the integral itself.

Midpoint Method

The midpoint method is a rectangular approximation technique used to estimate the value of a definite integral. It involves calculating the function's value at the midpoint of the interval and multiplying it by the interval's width.

Here's how to use the midpoint method:

• First, determine the midpoint of the interval \([a, b]\). For our example, this is \( x_{mid} = \frac{10 + 20}{2} = 15 \).
• Next, evaluate the function at this midpoint. Calculate \( f(15) = \left(1 + \frac{1}{15}\right)^5 \), which is approximately equivalent to \( 1.379 \).
• Finally, multiply this value by the interval's width, which is \( b - a = 20 - 10 = 10 \) in this scenario, resulting in the estimated integral value of \( 13.79 \).

The midpoint method provides a quick approximation and is particularly useful for functions with smooth curves over the interval.

Definite Integral

A definite integral is a fundamental concept in calculus, representing the accumulation of quantities, or the area under a curve, within a specific interval. It is expressed as \( \int_{a}^{b} f(x) \, dx \), where \( a \) and \( b \) are the bounds of the interval.

For example, \( \int_{10}^{20}\left(1+\frac{1}{x}\right)^{5} d x \) is a definite integral with:

• \( a = 10 \) and \( b = 20 \)
• \( f(x) = \left(1 + \frac{1}{x}\right)^{5} \)

The process involves understanding the function's behavior over \([10, 20]\) and calculating the equivalent area beneath its graph. Techniques like the midpoint method provide a way to estimate the value of such integrals when analytical solutions are complex or not feasible.

Function Plotting

Function plotting involves creating a visual representation of a mathematical function to better comprehend its shape and behavior over a set interval. This step is crucial in integral estimation, as it provides insights into how the function behaves between the bounds. By visualizing the function \( f(x) = \left(1+\frac{1}{x}\right)^5 \) from \([10, 20]\), we gain:

• A clear view of changes in the function's slope and curvature.
• Information to guide the estimation of regions under the curve where the integral is calculated.

Function plotting requires understanding calculators or graphing software, ensuring you adjust the settings to reflect the interval of interest accurately. This visualization helps verify the accuracy of the midpoint method or any numerical technique used for estimating definite integrals.