Problem 33 - Calculus Early Transcendentals

Question

Complete the following steps for the given function, interval, and value of \(n\) a. Sketch the graph of the function on the given interval. b. Calculate \(\Delta x\) and the grid points \(x_{0}, x_{1}, \ldots, x_{n}\) c. Illustrate the midpoint Riemann sum by sketching the appropriate rectangles. d. Calculate the midpoint Riemann sum. $$f(x)=\frac{1}{x} \text { on }[1,6] ; n=5$$

Step-by-Step Solution

Answer

Answer: The approximate midpoint Riemann sum is 1.944.

Step 1: Sketch the graph of the function on [1, 6]

The given function is \(f(x) = \frac{1}{x}\). On the interval [1, 6], this is a continuous, decreasing function. You can sketch the function with the understanding that it starts at the point (1, 1) and asymptotically approaches the x-axis as x increases.

Step 2: Calculate \(\Delta x\) and the grid points \(x_0, x_1, \ldots, x_n\)

To calculate \(\Delta x\), we use the formula \(\Delta x = \frac{b - a}{n}\), where 'a' and 'b' are the endpoints of the interval, and 'n' is the number of subintervals. In this case, \(a=1\), \(b=6\), and \(n=5\). So \(\Delta x=\frac{6-1}{5}=1\). Next, we need to find the grid points: \(x_0=1\), \(x_1=2\), \(x_2=3\), \(x_3=4\), \(x_4=5\), \(x_5=6\)

Step 3: Illustrate the midpoint Riemann sum by sketching the appropriate rectangles

To illustrate the midpoint Riemann sum, we need to find the midpoints between each grid point. Since \(\Delta x = 1\), the midpoint between each interval can be found by adding 0.5 to each grid point (except for the last one). Midpoints: \(x_{0.5}=1.5\), \(x_{1.5}=2.5\), \(x_{2.5}=3.5\), \(x_{3.5}=4.5\), \(x_{4.5}=5.5\) Now, sketch the rectangles using the midpoints on the function, which are the height of each respective rectangle, with each rectangle having a width of \(\Delta x=1\).

Step 4: Calculate the midpoint Riemann sum

To calculate the midpoint Riemann sum, we evaluate the function at the midpoint of each subinterval, then multiply the result by \(\Delta x\) and add them up: Riemann sum: \(\Delta x\left[f(x_{0.5})+f(x_{1.5})+f(x_{2.5})+f(x_{3.5})+f(x_{4.5})\right]\) Now, evaluate the function at each midpoint: \(f(x_{0.5})=\frac{1}{1.5}\), \(f(x_{1.5})=\frac{1}{2.5}\), \(f(x_{2.5})=\frac{1}{3.5}\), \(f(x_{3.5})=\frac{1}{4.5}\), \(f(x_{4.5})=\frac{1}{5.5}\) Riemann sum: \(1\left[\frac{1}{1.5}+\frac{1}{2.5}+\frac{1}{3.5}+\frac{1}{4.5}+\frac{1}{5.5}\right]=\frac{1}{1.5}+\frac{1}{2.5}+\frac{1}{3.5}+\frac{1}{4.5}+\frac{1}{5.5}\approx 1.944\) So, the midpoint Riemann sum of the given function on the interval [1, 6] with n=5 is approximately 1.944.

Key Concepts

Midpoint Rule

The Midpoint Rule is a useful technique for estimating the area under a curve, which gives an approximation of the definite integral. It utilizes the midpoints of subintervals to sample the values of the function. By using these midpoints, we create rectangles whose areas approximate the area under the curve.
The essential steps in the Midpoint Rule include:

• Dividing the given interval into equal subintervals.
• Identifying the midpoint of each subinterval.
• Calculating the function's value at each midpoint.
• Constructing rectangles using these function values as the heights and the subintervals widths as the bases.
• Summing up the areas of all these rectangles.

The Midpoint Rule provides a better approximation than other methods like the Left or Right Riemann Sum because it balances the overestimation and underestimation across the subintervals.

Interval Partition

An interval partition is a set of points that divides a given interval into smaller, non-overlapping subintervals. In this exercise, the interval \[1, 6\] is divided into 5 subintervals.
These partitions are crucial because they help define where we will evaluate the function. For a partition into \(n\) subintervals, you calculate the width \(\Delta x\) of each subinterval using the formula:

• \(\Delta x = \frac{b-a}{n}\)

where \(a\) and \(b\) are the lower and upper bounds of the interval, respectively.

The grid points \(x_0, x_1, \ldots, x_n\) at this partition consequently represent the ends of each subinterval. This setup is essential for structured and accurate numerical integration techniques like the Midpoint Rule.

Definite Integral

The definite integral of a function over an interval gives the total area under the curve between two points, typically denoted as \( \int_a^b f(x) \, dx \).
In this exercise, the interval \[1, 6\] represents the limits of integration for the function \(f(x) = \frac{1}{x}\).
Consider the integration to determine the exact area under the curve from \(x=1\) to \(x=6\).
The definite integral encompasses the concept of accumulation, providing precise values as opposed to the approximation methods.
For functions where algebraic integration isn't feasible, numerical integration techniques like the Midpoint Rule offer practical solutions.

Numerical Integration

Numerical integration refers to methods used to approximate the value of a definite integral when it's difficult or impossible to discover analytically. This is often necessary for complex functions or when the antiderivative does not exist.
Among various techniques, numerical integration includes methods like:

• The Midpoint Rule
• Trapezoidal Rule
• Simpson's Rule

These methods, including the Midpoint Rule from our exercise, rely on breaking up the integration interval into smaller segments, computing the function's value at specific points, and summing up contributions from each segment.
This allows us to estimate areas with desired precision and is crucial in many fields such as physics, engineering, and finance.