Problem 11 - A Graphical Approach to Precalculus with Limits

Question

Approximate the area under the graph of \(f(x)\) and above the \(x\) -axis, using each of the following methods with \(n=4\). (a) Use left endpoints. (b) Use right endpoints. (c) Average the answers in parts ( \(a\) ) and ( \(b\) ). (d) Use midpoints. $$f(x)=\frac{1}{x} \text { from } x=1 \text { to } x=5$$

Step-by-Step Solution

Answer

Left: 2.0833, Right: 1.2833, Average: 1.6833, Midpoints: 1.5746.

Step 1: Determine the interval width (Δx)

The function is given from \(x = 1\) to \(x = 5\). Since \(n = 4\), we divide the interval into 4 subintervals. The width of each subinterval is given by \(\Delta x = \frac{b-a}{n} = \frac{5-1}{4} = 1\).

Step 2: Approximate using left endpoints

For left endpoints, evaluate the function \(f(x)\) at the left end of each subinterval. The left endpoints are \(x = 1, 2, 3, 4\). Calculate the sum: \(\Delta x [f(1) + f(2) + f(3) + f(4)] = 1\left(\frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4}\right) = 1 + 0.5 + 0.3333 + 0.25 = 2.0833\).

Step 3: Approximate using right endpoints

For right endpoints, evaluate the function \(f(x)\) at the right end of each subinterval. The right endpoints are \(x = 2, 3, 4, 5\). Calculate the sum: \(\Delta x [f(2) + f(3) + f(4) + f(5)] = 1\left(\frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5}\right) = 0.5 + 0.3333 + 0.25 + 0.2 = 1.2833\).

Step 4: Average the left and right endpoint approximations

Average the results from parts (a) and (b): \(\frac{2.0833 + 1.2833}{2} = 1.6833\).

Step 5: Approximate using midpoints

For midpoints, evaluate the function \(f(x)\) at the midpoint of each subinterval. The midpoints are \(x = 1.5, 2.5, 3.5, 4.5\). Calculate the sum: \(\Delta x [f(1.5) + f(2.5) + f(3.5) + f(4.5)] = 1\left(\frac{1}{1.5} + \frac{1}{2.5} + \frac{1}{3.5} + \frac{1}{4.5}\right) \approx 0.6667 + 0.4 + 0.2857 + 0.2222 = 1.5746\).

Key Concepts

Riemann Sums

Riemann Sums are a fundamental concept in calculus used to approximate the area under a curve. Imagine dividing the area into several small rectangles, and finding the height of each rectangle by evaluating the function at a specific point within each subinterval. The sum of the areas of these rectangles gives us an approximation of the total area under the curve. By increasing the number of rectangles, or subintervals, we make this approximation more precise.

There are different types of Riemann Sums, including Left Endpoint, Right Endpoint, and Midpoint approximations. The choice of point within each subinterval influences the accuracy of the estimate.

• Left Endpoint Riemann Sum: Use the left endpoints of each subinterval for the height of the rectangles.
• Right Endpoint Riemann Sum: Use the right endpoints of the subintervals.
• Midpoint Riemann Sum: Use the midpoints of each subinterval for the height.

These methods are essential tools for understanding the fundamentals of integration and for solving practical problems where exact integration is difficult.

Left Endpoint Approximation

The Left Endpoint Approximation is a type of Riemann Sum where the height of each rectangle is determined by the value of the function at the left endpoint of each subinterval. To understand this, envision cutting the area under a curve into pieces, like slicing a loaf of bread.

In this approach, you take each slice starting from the left end. The left endpoint refers to the first x-value in each subinterval. For example, if we're evaluating the function from x = 1 to x = 5 with 4 subintervals, the left endpoints would be at x = 1, 2, 3, and 4. The width ( Δx) of each rectangle plays a crucial role, as it impacts the overall sum of the rectangular areas.

• Evaluate the function at each left endpoint: (f(1), f(2), f(3), f(4)).
• Multiply each evaluated function by the width ( Δx): ( Δx [f(1) + f(2) + f(3) + f(4)]).
• Sum these values to approximate the area: (2.0833).

Using the Left Endpoint Riemann Sum helps give an upper or lower bound of the actual area, depending on if the function is increasing or decreasing respectively.

Right Endpoint Approximation

The Right Endpoint Approximation is slightly different from the Left Endpoint method by using the rightmost x-value of each subinterval to determine the height of the rectangles.

This method can be visualized by considering each rectangle to start from the rightmost point, like reading a book from its back cover to the front. For an interval from x = 1 to x = 5 divided into 4 parts, the right endpoints are x = 2, 3, 4, and 5. The formula for this approximation is similar to the left endpoint but uses different x-values:

• Evaluate the function at each right endpoint: (f(2), f(3), f(4), f(5)).
• Calculate the products using the interval width ( Δx): ( Δx [f(2) + f(3) + f(4) + f(5)]).
• Add these values to get the right endpoint approximation of the area: (1.2833).

The usage of right endpoints often provides a differential error when compared to left endpoints, depending on whether the function is increasing or decreasing. Both offer clear visual cues on whether the actual integral might be larger or smaller than the approximation.

Midpoint Rule

The Midpoint Rule offers another Riemann Sum approach, providing often more accurate approximations than the Left or Right Endpoint methods. With this method, the height of each rectangle is determined by the value of the function at the midpoint of each subinterval.

Picture this as evaluating the function right in the center of each subinterval. In our earlier example between x = 1 and x = 5, midpoints might be situated at x = 1.5, 2.5, 3.5, and 4.5. Here's how the Midpoint Rule process unfolds:

• Evaluate the function at each midpoint: (f(1.5), f(2.5), f(3.5), f(4.5)).
• Multiply these values by the interval width: ( Δx [f(1.5) + f(2.5) + f(3.5) + f(4.5)]).
• Sum the products to estimate the area under the curve: (1.5746).

The Midpoint Rule often results in a closer approximation because it balances out the errors typically encountered in the Left and Right Endpoint methods. This makes it a practical choice in computational integration scenarios.