Approximating Areas: Left and Right Endpoint Methods

Learning Objectives

• Identify the need for rectangular approximations to find the area under a curve.
• Calculate the interval width, $\Delta x, for a given partition of size n$.
• Formulate and compute left-endpoint ($L_n) and right-endpoint (R_n$) approximations.
• Explain why increasing the number of rectangles ($n$) improves the area estimate.

Key Terms & Glossary

• Partition: The division of an interval $[a, b]$ into smaller, non-overlapping subintervals.
• Subinterval Width ($\Delta x$): The constant horizontal length of each individual rectangle in the approximation.
• Left-Endpoint Approximation ($L_n$): An area estimate where rectangle heights are determined by the function's value at the left edge of each subinterval.
• Right-Endpoint Approximation ($R_n$): An area estimate where rectangle heights are determined by the function's value at the right edge of each subinterval.

The "Big Idea"

Finding the exact area under a complex curve directly is nearly impossible. However, we can slice that complex area into simple geometric shapes—like rectangles—whose areas are easy to calculate ($A = \text{width} \times \text{height}$). By summing these rectangular areas, we obtain a reasonable estimate of the total curved region. The foundational "Big Idea" of calculus is that as we divide the region into smaller and smaller slices (letting the number of rectangles $n$ grow larger and larger), our estimate becomes increasingly accurate, eventually converging on the exact true area.

[!IMPORTANT] Increasing $n$ makes the rectangles thinner. This allows them to "hug" the true shape of the curve more precisely, minimizing the "wasted" or "over-estimated" empty space!

Formula / Concept Box

Hierarchical Outline

• 1. Setting Up the Approximation
  • Defining the bounds: Identify the interval starting point $a$ and ending point $b$.
  • Slicing the area: Choose $n$, the number of equal rectangles to place under the curve.
  • Calculating width: Use $\Delta x = (b-a)/n$ for the horizontal dimension of every slice.
• 2. Choosing the Evaluation Points
  • Left-Endpoints ($L_n$): Evaluate the function at the start of each subinterval to set height.
  • Right-Endpoints ($R_n$): Evaluate the function at the end of each subinterval to set height.
• 3. Improving Accuracy
  • Small $n$ (e.g., $n=4$): Provides a rough, blocky estimate with significant error.
  • Large $n$ (e.g., $n=32$): Rectangles become thin, fitting the curve much more precisely.
  • Infinite limit: As $n \to \infty, the discrepancy between L_n$ and $R_n$ shrinks to zero.

Visual Anchors

1. Flow of the Approximation Algorithm

2. Right-Endpoint Approximation ($n=3$)

Definition-Example Pairs

Worked Examples

▶Example 1: Calculating $R_4$ for a basic quadratic function

Problem: Approximate the area under $f(x) = x^2$ on the interval $[0, 2]$ using $n=4$ rectangles and right endpoints.

Step 1: Find subinterval width ($\Delta x$) $\Delta x = \frac{b - a}{n} = \frac{2 - 0}{4} = 0.5$

Step 2: Identify the right endpoints ($x_i$) The subintervals are [0, 0.5], [0.5, 1], [1, 1.5], [1.5, 2]. The right endpoints are: $0.5, 1, 1.5, and 2$.

Step 3: Evaluate function at these endpoints

• $f(0.5) = (0.5)^2 = 0.25$
• $f(1) = (1)^2 = 1.00$
• $f(1.5) = (1.5)^2 = 2.25$
• $f(2) = (2)^2 = 4.00$

Step 4: Multiply by $\Delta x$ and sum $R_4 = [f(0.5) + f(1) + f(1.5) + f(2)] \times \Delta x$ $R_4 = [0.25 + 1 + 2.25 + 4] \times 0.5 = 7.5 \times 0.5 = 3.75$

[!NOTE] The right-endpoint approximation of the area under this curve is exactly 3.75 square units.

▶Example 2: Calculating $L_4$ for the same function

Problem: Approximate the area under $f(x) = x^2$ on $[0, 2]$ using $n=4$ rectangles and left endpoints.

Step 1: Identify the left endpoints With $\Delta x = 0.5$, our subintervals are the same. The left endpoints are: $0, 0.5, 1, \text{ and } 1.5$.

Step 2: Evaluate function at these endpoints

• $f(0) = 0$
• $f(0.5) = 0.25$
• $f(1) = 1$
• $f(1.5) = 2.25$

Step 3: Multiply by $\Delta x$ and sum $L_4 = [f(0) + f(0.5) + f(1) + f(1.5)] \times \Delta x$ $L_4 = [0 + 0.25 + 1 + 2.25] \times 0.5 = 3.5 \times 0.5 = 1.75$

[!TIP] Notice how different $L_4$ (1.75) and $R_4 (3.75) are with small n$. As $n increases to 32, 100, or \infty$, these two estimated values will converge and reveal the true area!

Checkpoint Questions

1. If an interval is $[1, 5]$ and $n=8$, what is the width of each subinterval?
2. When computing a left-endpoint approximation ($L_n), do you ever evaluate the function at the very last point b of the entire interval [a, b]$? Why or why not?
3. According to the "Big Idea" of Riemann sums, what graphical change occurs when you increase the number of rectangles from $n=4$ to $n=32$?
4. How do you find the area of a single rectangular slice within a partition using mathematical notation?