Unit 6.2 – Approximating Areas with Riemann Sums

AP® Calculus AB & BC | The Foundation of Integration

Why This Matters: Riemann sums are the foundation of integration! They provide a systematic way to approximate the area under a curve by breaking it into rectangles (or trapezoids). This seemingly simple idea is incredibly powerful—it's how we measure everything from distances traveled to work done by forces to total accumulated change. Understanding Riemann sums deeply prepares you for the Fundamental Theorem of Calculus and gives you practical tools for numerical approximation when exact integration is impossible. This is one of the most important concepts in all of calculus!

🎯 The Fundamental Setup

RIEMANN SUM DEFINITION

A Riemann sum approximates the area under a curve \(y = f(x)\) from \(x = a\) to \(x = b\) by dividing the interval into \(n\) subintervals and summing the areas of rectangles.

General Form:

\[ \text{Riemann Sum} = \sum_{i=1}^{n} f(x_i^*) \Delta x \]

where:

\(n\) = number of subintervals (rectangles)
\(\Delta x = \frac{b-a}{n}\) = width of each subinterval
\(x_i^*\) = sample point in the \(i\)-th subinterval
\(f(x_i^*)\) = height of rectangle

🔑 The Key Formula - Width of Subintervals:

\[ \Delta x = \frac{b - a}{n} \]

This is THE most important formula to remember—you'll use it in EVERY Riemann sum problem!

📊 The Four Main Types of Riemann Sums

Complete Riemann Sum Formulas

1. LEFT Riemann Sum (LRS or LRAM)

Uses the left endpoint of each subinterval as the height.

\[ L_n = \sum_{i=0}^{n-1} f(x_i) \Delta x \]

\[ = \Delta x \left[ f(x_0) + f(x_1) + f(x_2) + \cdots + f(x_{n-1}) \right] \]

Partition points: \(x_i = a + i \cdot \Delta x\) for \(i = 0, 1, 2, \ldots, n-1\)

2. RIGHT Riemann Sum (RRS or RRAM)

Uses the right endpoint of each subinterval as the height.

\[ R_n = \sum_{i=1}^{n} f(x_i) \Delta x \]

\[ = \Delta x \left[ f(x_1) + f(x_2) + f(x_3) + \cdots + f(x_n) \right] \]

Partition points: \(x_i = a + i \cdot \Delta x\) for \(i = 1, 2, 3, \ldots, n\)

3. MIDPOINT Riemann Sum (MRS or MRAM)

Uses the midpoint of each subinterval as the height.

\[ M_n = \sum_{i=1}^{n} f(m_i) \Delta x \]

\[ = \Delta x \left[ f(m_1) + f(m_2) + f(m_3) + \cdots + f(m_n) \right] \]

Midpoints: \(m_i = \frac{x_{i-1} + x_i}{2} = a + \left(i - \frac{1}{2}\right) \Delta x\)

4. TRAPEZOIDAL Rule (Not technically Riemann sum, but related)

Uses trapezoids instead of rectangles (averages left and right).

\[ T_n = \frac{\Delta x}{2} \left[ f(x_0) + 2f(x_1) + 2f(x_2) + \cdots + 2f(x_{n-1}) + f(x_n) \right] \]

Pattern: First + 2(middle terms) + Last, all divided by 2

📈 Over-estimation vs Under-estimation

Critical Rules for Monotonic Functions:

When Riemann Sums Over/Under Estimate
Function Type	Left Riemann Sum	Right Riemann Sum	Midpoint/Trap
Increasing ↗	UNDER-estimates	OVER-estimates	More accurate
Decreasing ↘	OVER-estimates	UNDER-estimates	More accurate
Concave Up ∪	Under (trap)	Under (trap)	Trap: under; Mid: varies
Concave Down ∩	Over (trap)	Over (trap)	Trap: over; Mid: varies

💡 Memory Trick:

Increasing function: Left rectangles miss the increase → UNDER; Right catches increase → OVER
Decreasing function: Left rectangles catch extra → OVER; Right misses → UNDER
Visual check: Draw the function and rectangles—do rectangles include area outside the curve?

📋 Step-by-Step Procedure

Universal Riemann Sum Process:

Identify the interval: \([a, b]\)
Count subintervals: \(n\) (given in problem)
Calculate \(\Delta x\): \(\Delta x = \frac{b-a}{n}\)
Find partition points:
- \(x_0 = a\)
- \(x_i = a + i \cdot \Delta x\) for \(i = 1, 2, \ldots, n\)
Choose sample points:
- Left: Use \(x_0, x_1, \ldots, x_{n-1}\)
- Right: Use \(x_1, x_2, \ldots, x_n\)
- Midpoint: Use \(m_i = \frac{x_{i-1} + x_i}{2}\)
Evaluate function: Find \(f(x_i^*)\) at each sample point
Sum and multiply: \(\sum f(x_i^*) \times \Delta x\)
Simplify: Calculate final numerical answer

📖 Comprehensive Worked Examples

Example 1: All Four Methods

Problem: Approximate \(\int_1^5 x^2 \, dx\) using \(n = 4\) subintervals with:
(a) Left Riemann Sum
(b) Right Riemann Sum
(c) Midpoint Riemann Sum
(d) Trapezoidal Rule

Solution:

Step 1-3: Basic Setup

Interval: \([1, 5]\), so \(a = 1, b = 5\)
Number of subintervals: \(n = 4\)
Width: \(\Delta x = \frac{5-1}{4} = \frac{4}{4} = 1\)

Step 4: Partition Points

\[ x_0 = 1, \quad x_1 = 2, \quad x_2 = 3, \quad x_3 = 4, \quad x_4 = 5 \]

Part (a): Left Riemann Sum

Use: \(x_0, x_1, x_2, x_3\)

\[ L_4 = \Delta x [f(1) + f(2) + f(3) + f(4)] \]

\[ = 1[1^2 + 2^2 + 3^2 + 4^2] = 1[1 + 4 + 9 + 16] = 30 \]

Part (b): Right Riemann Sum

Use: \(x_1, x_2, x_3, x_4\)

\[ R_4 = \Delta x [f(2) + f(3) + f(4) + f(5)] \]

\[ = 1[2^2 + 3^2 + 4^2 + 5^2] = 1[4 + 9 + 16 + 25] = 54 \]

Part (c): Midpoint Riemann Sum

Midpoints: \(m_1 = 1.5, m_2 = 2.5, m_3 = 3.5, m_4 = 4.5\)

\[ M_4 = \Delta x [f(1.5) + f(2.5) + f(3.5) + f(4.5)] \]

\[ = 1[2.25 + 6.25 + 12.25 + 20.25] = 41 \]

Part (d): Trapezoidal Rule

\[ T_4 = \frac{\Delta x}{2} [f(1) + 2f(2) + 2f(3) + 2f(4) + f(5)] \]

\[ = \frac{1}{2}[1 + 2(4) + 2(9) + 2(16) + 25] \]

\[ = \frac{1}{2}[1 + 8 + 18 + 32 + 25] = \frac{84}{2} = 42 \]

Answers: (a) \(L_4 = 30\) | (b) \(R_4 = 54\) | (c) \(M_4 = 41\) | (d) \(T_4 = 42\)
Note: Exact value is \(\frac{124}{3} \approx 41.33\). Midpoint is closest!
Since \(f(x) = x^2\) is increasing, Left UNDER-estimates, Right OVER-estimates ✓

Example 2: Using a Table of Values

Problem: Use the data to approximate \(\int_0^8 f(x) \, dx\) with the Trapezoidal Rule.

\(x\)	0	2	4	6	8
\(f(x)\)	3	7	11	9	5

Solution:

Step 1-2: Identify setup

Interval: \([0, 8]\)
Values given at: 0, 2, 4, 6, 8 → \(n = 4\) subintervals
\(\Delta x = 2\)

Step 3: Apply Trapezoidal Rule

\[ T_4 = \frac{\Delta x}{2}[f(0) + 2f(2) + 2f(4) + 2f(6) + f(8)] \]

\[ = \frac{2}{2}[3 + 2(7) + 2(11) + 2(9) + 5] \]

\[ = 1[3 + 14 + 22 + 18 + 5] = 62 \]

Answer: \(\int_0^8 f(x) \, dx \approx 62\)

Example 3: Determining Over/Under Estimation

Problem: For \(f(x) = e^{-x}\) on \([0, 3]\), determine whether the Left Riemann Sum with \(n = 6\) over-estimates or under-estimates the actual integral.

Solution:

Step 1: Analyze the function

\(f(x) = e^{-x}\) is an exponential decay function.

\(f'(x) = -e^{-x} < 0\) for all \(x\)
Therefore, \(f\) is DECREASING on \([0, 3]\)

Step 2: Apply the rule

For a decreasing function:

Left Riemann Sum uses higher values (left endpoints)
This captures more area than actually exists
Therefore: OVER-estimates

Answer: The Left Riemann Sum OVER-estimates because \(f(x) = e^{-x}\) is decreasing.

Example 4: Summation Notation

Problem: Express the Right Riemann Sum for \(\int_2^{10} (3x + 1) \, dx\) with \(n = 4\) using summation notation, then evaluate.

Solution:

Step 1: Setup

\(a = 2, b = 10, n = 4\)
\(\Delta x = \frac{10-2}{4} = 2\)
\(x_i = 2 + 2i\) for \(i = 1, 2, 3, 4\)

Step 2: Write summation notation

\[ R_4 = \sum_{i=1}^{4} f(x_i) \Delta x = \sum_{i=1}^{4} f(2 + 2i) \cdot 2 \]

\[ = \sum_{i=1}^{4} [3(2 + 2i) + 1] \cdot 2 = \sum_{i=1}^{4} [6 + 6i + 1] \cdot 2 \]

\[ = \sum_{i=1}^{4} (7 + 6i) \cdot 2 = 2 \sum_{i=1}^{4} (7 + 6i) \]

Step 3: Evaluate

\(i = 1: 7 + 6(1) = 13\)
\(i = 2: 7 + 6(2) = 19\)
\(i = 3: 7 + 6(3) = 25\)
\(i = 4: 7 + 6(4) = 31\)

\[ R_4 = 2(13 + 19 + 25 + 31) = 2(88) = 176 \]

Answer: \(R_4 = 2\sum_{i=1}^{4}(7+6i) = 176\)

🖩 Calculator Techniques

✅ Using Your Graphing Calculator:

For TI-84 Family:

Store function: \(Y_1 = f(x)\)
Create list of x-values: \(\text{seq}(a + i \cdot \Delta x, i, 0, n-1)\) → L1 (for Left)
Calculate function values: \(Y_1(\text{L1})\) → L2
Sum: \(\Delta x \cdot \text{sum}(\text{L2})\)

📝 Important: On AP® exams, if you use a calculator, you must show:

The Riemann sum setup (formula)
Value of \(\Delta x\)
Which method you're using
The final numerical answer (rounded appropriately)

💡 Essential Tips & Tricks

🔥 Pro Tips:

Always calculate \(\Delta x\) first: It's needed for everything!
Count carefully: Left uses \(n\) values (first \(n\)), Right uses \(n\) values (last \(n\))
Trapezoidal pattern: First + 2(middles) + Last, divided by 2
Check reasonableness: More rectangles → better approximation
For tables: Trapezoidal Rule is easiest and most accurate
Midpoint usually best: For smooth functions, midpoint is most accurate
Don't round \(\Delta x\): Keep exact fraction if possible
Organization is key: Make a table of \(x_i\) and \(f(x_i)\) values

🎯 Quick Reference for Over/Under Estimation:

Function Behavior	Left Sum	Right Sum
Increasing	UNDER ↓	OVER ↑
Decreasing	OVER ↑	UNDER ↓

❌ Common Mistakes to Avoid

Mistake 1: Forgetting to multiply by \(\Delta x\) (most common error!)
Mistake 2: Using \(n+1\) values instead of \(n\) values
Mistake 3: Mixing up left and right endpoints
Mistake 4: In Trapezoidal Rule, forgetting the \(\frac{1}{2}\) factor
Mistake 5: In Trapezoidal Rule, not doubling the middle terms
Mistake 6: Calculating wrong midpoints (should be average of consecutive points)
Mistake 7: Using \(n\) as the interval length instead of \(\Delta x\)
Mistake 8: Rounding \(\Delta x\) too early (causes cumulative error)
Mistake 9: For tables, not recognizing the spacing between x-values
Mistake 10: Confusing over/under estimation rules
Mistake 11: Not checking if function is actually monotonic before claiming over/under
Mistake 12: Arithmetic errors when adding many terms (use calculator!)

📝 Practice Problems

Set A: Basic Computations

Approximate \(\int_0^4 x^3 \, dx\) using Left Riemann Sum with \(n = 4\)
Approximate \(\int_1^3 \frac{1}{x} \, dx\) using Right Riemann Sum with \(n = 4\)
Use Midpoint Riemann Sum with \(n = 3\) to approximate \(\int_0^3 (x^2 + 1) \, dx\)

Answers:

\(L_4 = 1[0 + 1 + 8 + 27] = 36\)
\(R_4 = 0.5[\frac{1}{1.5} + \frac{1}{2} + \frac{1}{2.5} + \frac{1}{3}] \approx 0.878\)
\(M_3 = 1[1.25 + 4.25 + 9.25] = 14.75\)

Set B: Table Problems

Use Trapezoidal Rule to approximate the integral given:

\(x\) 0 3 6 9

\(f(x)\) 2 5 7 4

Answer:

\(T_3 = \frac{3}{2}[2 + 2(5) + 2(7) + 4] = \frac{3}{2}(30) = 45\)

Set C: Conceptual

If \(f(x) = \sin(x)\) on \([0, \pi]\), does Right Riemann Sum with \(n=6\) over or under-estimate?
True or False: For any function, increasing \(n\) always improves the approximation.

Answers:

Over-estimate (function decreasing on \((\frac{\pi}{2}, \pi)\), but need to analyze carefully)
True (as \(n \to \infty\), all methods converge to exact value)

✏️ AP® Exam Success Tips

What AP® Graders Look For:

Setup shown: Write the formula before calculating
\(\Delta x\) calculated: Show \(\Delta x = \frac{b-a}{n}\)
Sample points identified: List x-values used
Function evaluations: Show \(f(x_i)\) values
Summation clear: Show the sum before final answer
Units included: If context problem
Proper notation: \(L_n, R_n, M_n, T_n\)
Calculator note: If used, state which calculation

💯 Maximizing Your Score:

Show work: Even with calculator—show setup first
Be organized: Table format for x-values and f(x) values
Label clearly: "Left Riemann Sum", "Trapezoidal Rule", etc.
Check arithmetic: Use calculator to verify
Round appropriately: 3 decimal places unless specified

⚡ Ultimate Quick Reference

Complete Riemann Sum Reference
Type	Sample Points	Formula Pattern
Left	\(x_0, x_1, \ldots, x_{n-1}\)	\(\Delta x[f(x_0) + f(x_1) + \cdots + f(x_{n-1})]\)
Right	\(x_1, x_2, \ldots, x_n\)	\(\Delta x[f(x_1) + f(x_2) + \cdots + f(x_n)]\)
Midpoint	\(m_1, m_2, \ldots, m_n\)	\(\Delta x[f(m_1) + f(m_2) + \cdots + f(m_n)]\)
Trapezoidal	All \(x_0\) through \(x_n\)	\(\frac{\Delta x}{2}[f(x_0) + 2f(x_1) + \cdots + 2f(x_{n-1}) + f(x_n)]\)

ALWAYS REMEMBER: \(\Delta x = \frac{b-a}{n}\)

Master Riemann Sums! These approximation methods are fundamental to understanding integration. The key formula is \(\Delta x = \frac{b-a}{n}\)—memorize it! The four main methods are: Left (uses first \(n\) points), Right (uses last \(n\) points), Midpoint (uses midpoints of subintervals), and Trapezoidal (first + 2×middle + last, divided by 2). For monotonic functions, remember: Left under-estimates increasing functions and over-estimates decreasing functions; Right does the opposite. As \(n \to \infty\), all methods converge to the exact integral value. Trapezoidal Rule is best for data tables. Always show your setup: calculate \(\Delta x\), list sample points, evaluate the function, then sum and multiply. Organization is crucial—make tables of values! Practice until the patterns become automatic, and always double-check that you multiplied by \(\Delta x\). These techniques appear frequently on AP® exams! 🎯✨

\(x\)	0	3	6	9
\(f(x)\)	2	5	7	4