Unit 6.1 – Exploring Accumulations of Change

AP® Calculus AB & BC | Introduction to Integration and Accumulation

Why This Matters: Welcome to Unit 6 - Integration! This is where we flip the script on derivatives. While derivatives measure instantaneous rate of change, integration measures accumulation of change over time. If velocity is the derivative of position, then position is the accumulation of velocity. If marginal cost is the derivative of total cost, then total cost is the accumulation of marginal cost. This fundamental concept—that accumulation "undoes" differentiation—is one of the most powerful ideas in all of mathematics. It's called the Fundamental Theorem of Calculus, and it revolutionizes how we solve real-world problems!

🎯 The Big Picture: From Rates to Totals

The Central Idea

If you know the RATE of change, you can find the TOTAL change by accumulation.

The Fundamental Relationship:

\[ \text{Total Change} = \text{Accumulation of Rate} \]

\[ F(b) - F(a) = \int_a^b F'(x) \, dx \]

In words: The change in quantity from \(a\) to \(b\) equals the accumulation of its rate of change.

Real-World Examples:

Rates and Accumulations
Rate of Change	Accumulation Gives	Interpretation
Velocity \(v(t)\)	Displacement	Total distance traveled
Flow rate \(r(t)\)	Total volume	Amount of fluid accumulated
Growth rate \(g(t)\)	Total growth	Change in population/size
Marginal cost \(C'(x)\)	Total cost increase	Additional cost for production
Power \(P(t)\)	Energy	Total energy consumed

📊 Accumulation as Area Under a Curve

GEOMETRIC INTERPRETATION

The accumulation of a function \(f(x)\) from \(x = a\) to \(x = b\) equals the signed area between the curve and the x-axis.

Area Interpretation:

If \(f(x) > 0\): Accumulation adds positive area (above x-axis)
If \(f(x) < 0\): Accumulation adds negative area (below x-axis)
Net accumulation: Sum of positive and negative areas

📝 Important Distinction:

Net change (signed area): Accounts for direction; can be negative
Total distance/amount: Uses absolute values; always positive
Example: If velocity alternates positive/negative, net displacement ≠ total distance

🔢 Approximating Accumulation: Riemann Sums

Riemann Sum Methods

To approximate accumulation, we divide the interval \([a, b]\) into \(n\) subintervals of width \(\Delta x = \frac{b-a}{n}\) and sum up rectangles.

General Riemann Sum:

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

where \(x_i^*\) is a sample point in the \(i\)-th subinterval

Left Riemann Sum (LRS):

\[ L_n = \sum_{i=0}^{n-1} f(x_i) \Delta x = \Delta x [f(x_0) + f(x_1) + \cdots + f(x_{n-1})] \]

Uses left endpoint of each subinterval

Right Riemann Sum (RRS):

\[ R_n = \sum_{i=1}^{n} f(x_i) \Delta x = \Delta x [f(x_1) + f(x_2) + \cdots + f(x_n)] \]

Uses right endpoint of each subinterval

Midpoint Riemann Sum (MRS):

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

where \(m_i = \frac{x_{i-1} + x_i}{2}\) is the midpoint of each subinterval

Trapezoidal Sum (TS):

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

Uses trapezoids instead of rectangles (average of left and right)

🔑 Key Properties of Riemann Sums:

As \(n \to \infty\): All Riemann sums approach the exact value (the definite integral)
For increasing functions: LRS underestimates, RRS overestimates
For decreasing functions: LRS overestimates, RRS underestimates
Midpoint and Trapezoidal: Generally more accurate than left/right
\(\Delta x\) smaller: Better approximation

📖 Comprehensive Worked Examples

Example 1: Computing Riemann Sums

Problem: Approximate \(\int_0^4 x^2 \, dx\) using 4 subintervals with:
(a) Left Riemann Sum
(b) Right Riemann Sum
(c) Midpoint Riemann Sum

Solution:

Setup:

Interval: \([0, 4]\)
Number of subintervals: \(n = 4\)
Width: \(\Delta x = \frac{4-0}{4} = 1\)
Partition points: \(x_0 = 0, x_1 = 1, x_2 = 2, x_3 = 3, x_4 = 4\)

Part (a): Left Riemann Sum

\[ L_4 = \Delta x [f(0) + f(1) + f(2) + f(3)] \]

\[ L_4 = 1[0^2 + 1^2 + 2^2 + 3^2] = 1[0 + 1 + 4 + 9] = 14 \]

Part (b): Right Riemann Sum

\[ R_4 = \Delta x [f(1) + f(2) + f(3) + f(4)] \]

\[ R_4 = 1[1^2 + 2^2 + 3^2 + 4^2] = 1[1 + 4 + 9 + 16] = 30 \]

Part (c): Midpoint Riemann Sum

Midpoints: \(m_1 = 0.5, m_2 = 1.5, m_3 = 2.5, m_4 = 3.5\)

\[ M_4 = \Delta x [f(0.5) + f(1.5) + f(2.5) + f(3.5)] \]

\[ M_4 = 1[0.25 + 2.25 + 6.25 + 12.25] = 21 \]

Answers: (a) \(L_4 = 14\) | (b) \(R_4 = 30\) | (c) \(M_4 = 21\)
Note: Exact value is \(\frac{64}{3} \approx 21.33\); midpoint is closest!

Example 2: Accumulation from a Table

Problem: A car's velocity \(v(t)\) (in ft/s) is recorded every 2 seconds:

\(t\) (seconds)	0	2	4	6	8
\(v(t)\) (ft/s)	10	15	18	20	22

Approximate the distance traveled from \(t = 0\) to \(t = 8\) using the Trapezoidal Rule.

Solution:

Setup:

\(\Delta t = 2\) seconds
\(n = 4\) subintervals

Trapezoidal Rule:

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

\[ T_4 = \frac{2}{2}[10 + 2(15) + 2(18) + 2(20) + 22] \]

\[ T_4 = 1[10 + 30 + 36 + 40 + 22] = 138 \text{ feet} \]

Answer: The car traveled approximately 138 feet from \(t = 0\) to \(t = 8\) seconds.

Example 3: Accumulation with Graph

Problem: A function \(f(t)\) represents the rate at which water flows into a tank (in gallons per minute). From a graph:

From \(t = 0\) to \(t = 3\): \(f(t)\) is constant at 5 gal/min
From \(t = 3\) to \(t = 6\): \(f(t)\) decreases linearly from 5 to 2 gal/min
From \(t = 6\) to \(t = 8\): \(f(t)\) is constant at 2 gal/min

How much water enters the tank from \(t = 0\) to \(t = 8\) minutes?

Solution:

Method: Calculate area under curve

Region 1: \([0, 3]\)

Rectangle: \(A_1 = 5 \times 3 = 15\) gallons

Region 2: \([3, 6]\)

Trapezoid: \(A_2 = \frac{1}{2}(5 + 2) \times 3 = \frac{21}{2} = 10.5\) gallons

Region 3: \([6, 8]\)

Rectangle: \(A_3 = 2 \times 2 = 4\) gallons

Total Accumulation:

\[ \text{Total} = A_1 + A_2 + A_3 = 15 + 10.5 + 4 = 29.5 \text{ gallons} \]

Answer: 29.5 gallons of water entered the tank.

📝 Notation and Terminology

Key Terms:

Definite Integral (Coming Soon):

\[ \int_a^b f(x) \, dx \]

\(\int\) = integral sign (elongated S for "sum")
\(a\) = lower limit of integration
\(b\) = upper limit of integration
\(f(x)\) = integrand (function being accumulated)
\(dx\) = indicates variable of integration

Accumulation Function:

\[ A(x) = \int_a^x f(t) \, dt \]

Accumulation from fixed point \(a\) to variable point \(x\)

💡 Tips, Tricks & Strategies

✅ Essential Accumulation Tips:

Think "total from rate": Accumulation reverses differentiation
Area interpretation: Always visualize as area under curve
Sign matters: Negative values subtract from accumulation
Units check: Rate units × time units = accumulation units
Riemann sum choice: More rectangles = better approximation
Trapezoidal is best: For equally spaced data, it's most accurate
Graph reading: Break complex regions into simple shapes
Calculator allowed: Use it for arithmetic, but show setup!

🔥 Quick Decision Guide:

When to Use Each Method
Situation	Best Method	Why
Data table given	Trapezoidal Rule	Works with discrete data
Function given	Any Riemann sum	Can evaluate at any point
Graph given	Geometric shapes	Triangles, rectangles, trapezoids
Need estimate	Midpoint or Trapezoidal	More accurate than left/right
Increasing function	Left = underestimate	Right = overestimate

❌ Common Mistakes to Avoid

Mistake 1: Forgetting to multiply by \(\Delta x\) in Riemann sums
Mistake 2: Using wrong endpoints (left vs. right vs. midpoint)
Mistake 3: Not accounting for negative areas below x-axis
Mistake 4: Confusing net change with total distance/amount
Mistake 5: Miscounting number of rectangles or trapezoids
Mistake 6: In trapezoidal rule, forgetting to divide by 2
Mistake 7: Not including first and last values in trapezoidal sum
Mistake 8: Mixing up rate (derivative) and total (accumulation)
Mistake 9: Wrong units in final answer
Mistake 10: Not showing setup before using calculator

📝 Practice Problems

Set A: Riemann Sums

Approximate \(\int_0^6 (2x+1) \, dx\) using \(n=3\) and Left Riemann Sum
Use Right Riemann Sum with \(n=4\) to approximate \(\int_1^5 \frac{1}{x} \, dx\)
For \(f(x) = x^2 - 2x\) on \([0, 4]\), find Midpoint Riemann Sum with \(n=4\)

Answers:

\(L_3 = 2[1 + 5 + 9] = 30\)
\(R_4 = 1[\frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5}] \approx 1.283\)
\(M_4 = 1[-0.75 + 0 + 1.25 + 4.75] = 5.25\)

Set B: Applied Problems

Water flows into tank at rate \(r(t) = 5 + 2t\) gal/min. How much water in first 4 minutes? (Use \(n=4\), Trap. Rule)
A particle's velocity is \(v(t) = 3t^2 - 6t\) m/s. Estimate displacement from \(t=0\) to \(t=3\) using \(n=3\), Right Riemann Sum.

Answers:

\(T_4 = 36\) gallons
\(R_3 = -3\) meters (3 meters in negative direction)

✏️ AP® Exam Success Tips

What AP® Graders Look For:

Clear setup: Show \(\Delta x\) and partition points
Riemann sum formula: Write complete expression before calculating
Work shown: Even if using calculator, show what you're computing
Proper notation: Use summation notation or expanded form correctly
Units included: State units in context problems
Interpretation: Explain what accumulation represents
Justification: State which method and why (if asked)

⚡ Quick Reference Card

Accumulation Quick Reference
Method	Formula	When to Use
Left Riemann	\(\sum_{i=0}^{n-1} f(x_i) \Delta x\)	Use left endpoints
Right Riemann	\(\sum_{i=1}^{n} f(x_i) \Delta x\)	Use right endpoints
Midpoint	\(\sum_{i=1}^{n} f(m_i) \Delta x\)	Use midpoints; more accurate
Trapezoidal	\(\frac{\Delta x}{2}[f(x_0) + 2\sum_{i=1}^{n-1}f(x_i) + f(x_n)]\)	Best for data tables
\(\Delta x\)	\(\frac{b-a}{n}\)	Subinterval width

Welcome to Integration! This unit introduces the fundamental concept that accumulation reverses differentiation. When you know the rate of change, you can find total change by accumulating over an interval. Geometrically, this is area under a curve. We approximate accumulation using Riemann sums: Left, Right, Midpoint, and Trapezoidal methods, each using rectangles or trapezoids of width \(\Delta x = \frac{b-a}{n}\). The Trapezoidal Rule is typically most accurate and works great with data tables. Remember: accumulation accounts for sign (positive and negative areas), and as \(n \to \infty\), all Riemann sums converge to the exact integral. Key applications include distance from velocity, total volume from flow rate, and total cost from marginal cost. Always multiply by \(\Delta x\), include units, and show your setup clearly on AP® exams. This is just the beginning—the Fundamental Theorem of Calculus will formalize these ideas and give us powerful tools for exact computation! 🎯✨