Accumulation Slides Accumulation Functions
Watching the Area Grow
The Filling Glass
Imagine you are filling a vase with water at a constant rate .
"As time passes, how does the volume of water change if the vase gets wider or narrower?"
The rate of change is the "input"
The total volume is the "accumulation"
The Formal Definition
An accumulation function \( g(x) \) represents the net signed area under the graph of \( f \) from a fixed point \( a \) to a variable point \( x \):
\[ g(x) = \int_{a}^{x} f(t) \, dt \]
\( a \)
Fixed Starting Point
\( x \)
Variable Upper Limit
\( t \)
Dummy Variable
Visualizing \( g(x) \)
a x f(t)
Key Behaviors:
1
If \( f(t) > 0 \), then \( g(x) \) is increasing as \( x \) moves right.
2
If \( f(t) < 0 \), then \( g(x) \) is decreasing as \( x \) moves right.
3
\( g(a) = 0 \) always.
Ready to Accumulate?
Open your "Filling the Void" worksheet. We are going to track how a specific function grows over time and see if we can spot a pattern in its derivative.
Think About
What happens to the area if the function stays constant?
Observe
What point makes the accumulation function reach its maximum value?
Filling the Void Worksheet Filling the Void
Accumulation Functions Exploration
Name:
Date:
Goal
Investigate the behavior of the accumulation function \( g(x) = \int_{0}^{x} f(t) \, dt \) for a given rate function \( f(t) \).
1 The Rate Function \( f(t) \)
f(t) t 2 4 6
Define \( g(x) = \int_{0}^{x} f(t) \, dt \)
Use the graph of \( f(t) \) above to fill out the table of values for the accumulation function.
2 Table Analysis
\( x \) Integral Calculation \( g(x) \) 0 Example: \(\int_{0}^{0} f(t)dt\) 0 1 2 3 4 6
3 Sketching the Accumulation
Plot the points from your table and sketch the graph of \( g(x) \) on the interval \([0, 6]\).
0123456
4 Deep Dive Questions
1. At what x-value does \( g(x) \) reach its maximum value? Why does this point occur there in relation to \( f(t) \)?
2. Look at the interval where \( f(t) \) is negative. What happens to the slope of your graph for \( g(x) \) on that same interval?
3. Based on your sketch, what would you guess is the relationship between the derivative \( g'(x) \) and the original function \( f(x) \)?
Accumulation Facilitation Guide Accumulation Facilitation Guide
Lesson 1: Introduction to Variable Upper Limits
Duration
55 Minutes
Prereqs
Basic Definite Integrals, Area Formulas
Materials
Slides, "Filling the Void" Worksheet
1. The Hook (10 mins)
Use the Vase Analogy from the slides. Ask students:
"If the vase is a cylinder (constant width), how does the height change?" (Linear)
"If the vase gets wider at the top, does the height grow faster or slower near the end?" (Slower)
Key Insight: The shape of the vase is the rate function \( f(t) \), and the height/volume is the accumulation \( g(x) \).
2. Guided Practice (15 mins)
Walk through Slide 3 and 4. Emphasize why we use \( t \) as the variable inside the integral and \( x \) as the bound.
Common Misconception
Students often confuse \( f(t) \) with \( g(x) \). Remind them: \( f(t) \) is the height of the curve; \( g(x) \) is the area accumulated so far.
Check for Understanding
Ask: "What is \( g(0) \) if we start at \( a=0 \)? Why?" (Answer: 0, because an integral with zero width has zero area).
3. Worksheet & Investigation (20 mins)
Have students work on "Filling the Void." While they work, circulate and ask these targeted questions:
Student observes... Ask them... \( f(t) \) is constant (0 to 2) "What type of function is \( g(x) \) here? Linear, Quadratic?" \( f(t) \) is zero "Is the area growing? What does that mean for the graph of \( g(x) \)? " Max value of \( g(x) \) "Does the max happen where \( f(t) \) is max, or where \( f(t) \) crosses the x-axis?"
4. Debrief (10 mins)
The final question on the worksheet is the bridge to Lesson 2. Conduct a quick poll:
g'(x)
is the same as
f(x)
This realization prepares them for the Fundamental Theorem of Calculus Part 1.
The Inverse Link Slides The Inverse Link
The Fundamental Theorem of Calculus - Part 1
What is FTC Part 1?
If \( f \) is continuous on \([a, b]\), then the function \( g(x) = \int_a^x f(t)dt \) is continuous on \([a, b]\) and differentiable on \((a, b)\), and:
\[ \frac{d}{dx} \left[ \int_{a}^{x} f(t) \, dt \right] = f(x) \]
In plain English: Differentiation and Integration are inverses!
A Visual Intuition
1. Area \( \int_a^x f(t)dt \) is the accumulation.
2. Derivative \( \frac{d}{dx} \) asks: "How fast is that area changing right now?"
3. At the exact moment \( x \), the rate of area growth is simply the height of the function \( f(x) \).
Imagine pushing a vertical line at speed 1. The rate at which the shaded area grows is equal to the height of that line.
x x + h Growth \(\approx f(x) \cdot h\)
What if the upper bound isn't just \( x \)?
If the upper bound is a function \( u(x) \), we must use the Chain Rule :
\[ \frac{d}{dx} \int_{a}^{g(x)} f(t) \, dt = f(g(x)) \cdot g'(x) \]
Example 1:
\[ \frac{d}{dx} \int_{1}^{x^2} \sin(t) \, dt \]
Answer: \( \sin(x^2) \cdot 2x \)
Example 2:
\[ \frac{d}{dx} \int_{x}^{5} t^3 \, dt \]
Answer: \( -x^3 \)
The Derivative Hunt
You are about to receive a series of complex-looking integrals. Your goal is to differentiate them using the FTC Part 1 shortcut.
Rule of Thumb:
"Plug it in, and multiply by the derivative of the bound."
Derivative Hunt Activity The Derivative Hunt
FTC Part 1 Discovery & Practice
Name: __________________________
The Standard Rule
\[ \frac{d}{dx} \int_{a}^{x} f(t) dt = f(x) \]
The Chain Rule Twist
\[ \frac{d}{dx} \int_{a}^{u(x)} f(t) dt = f(u(x)) \cdot u'(x) \]
01
Find \( F'(x) \) if:
\[ F(x) = \int_{3}^{x} \sqrt{t^2 + 1} \, dt \]
Solution area
02
Find \( \frac{dy}{dx} \) if:
\[ y = \int_{x}^{10} \frac{1}{t + \sin t} \, dt \]
Watch the bounds!
Solution area
03
Evaluate the derivative:
\[ \frac{d}{dx} \int_{2}^{\sin x} e^{t^2} \, dt \]
Solution area
04
Level Up Challenge:
\[ \frac{d}{dx} \int_{x^2}^{x^3} \ln(t) \, dt \]
Hint: Split the integral at a constant \( a \).
Solution area
The Inverse Paradox
If \( f(x) = \frac{d}{dx} \int_{a}^{x} f(t) dt \), does it matter what the constant \( a \) is? Does \( a = 0 \) give a different derivative than \( a = 100 \)? Explain your reasoning.
Derivative Hunt Key The Derivative Hunt (Key)
Teacher Answer Key
Problem 01:
\[ F(x) = \int_{3}^{x} \sqrt{t^2 + 1} \, dt \]
Answer: \( F'(x) = \sqrt{x^2 + 1} \)
Logic: Standard application. The upper bound is \( x \), so just replace \( t \) with \( x \).
Problem 02:
\[ y = \int_{x}^{10} \frac{1}{t + \sin t} \, dt \]
Answer: \( \frac{dy}{dx} = -\frac{1}{x + \sin x} \)
Logic: The bound is reversed. \( \int_{x}^{10} = -\int_{10}^{x} \). Don't forget the negative sign!
Problem 03:
\[ \frac{d}{dx} \int_{2}^{\sin x} e^{t^2} \, dt \]
Answer: \( e^{\sin^2 x} \cdot \cos x \)
Logic: Chain Rule. Plug in \( \sin x \), then multiply by the derivative of \( \sin x \), which is \( \cos x \).
Problem 04:
\[ \frac{d}{dx} \int_{x^2}^{x^3} \ln(t) \, dt \]
Answer: \( \ln(x^3) \cdot 3x^2 - \ln(x^2) \cdot 2x \)
Logic: Split at constant \( c \): \( \int_{x^2}^{c} + \int_{c}^{x^3} \). Apply chain rule to both terms (one will be negative due to reversed bounds).
The Inverse Paradox Explanation:
The constant \( a \) does NOT matter. When you differentiate the integral, the value \( a \) is essentially the "starting constant" that disappears during differentiation (just like the derivative of any constant is zero). Visually, changing \( a \) shifts the accumulation function up or down vertically, but its slope at any point \( x \) remains the same height as the original curve \( f(x) \).
Evaluation Shortcut Slides The Evaluation Shortcut
The Fundamental Theorem of Calculus - Part 2
The Power of Part 2
This theorem allows us to calculate exact area without needing complex limits or Riemann sums.
If \( F \) is any antiderivative of \( f \), then:
\[ \int_{a}^{b} f(x) \, dx = F(b) - F(a) \]
"Evaluate the antiderivative at the top, subtract the antiderivative at the bottom."
The Evaluation Protocol
01
Antiderive
Find any antiderivative \( F(x) \). You can ignore the \( +C \) because it will cancel out anyway!
02
Bracket
Use the evaluation bracket notation:
\[ [F(x)]_{a}^{b} \]
03
Subtract
Plugin the upper limit first, then the lower limit, and find the difference.
The Polynomial Challenge
Solve: \[ \int_{1}^{3} (x^2 + 1) \, dx \]
Antiderivative: \( \frac{x^3}{3} + x \)
Evaluate: \( [\frac{x^3}{3} + x]_{1}^{3} \)
Substitute: \( (\frac{27}{3} + 3) - (\frac{1}{3} + 1) \)
Result: \( 12 - \frac{4}{3} = \frac{32}{3} \approx 10.67 \)
Visual Verification
The area under the parabola is exactly \( \frac{32}{3} \).
Watch Your Signs!
The Subtraction Error
Always use parentheses!
Forgetting to distribute the negative is the #1 error in Calculus.
Continuity Check
The FTC only works if the function is continuous on the interval.
You can't evaluate \( \int_{-1}^{1} \frac{1}{x^2} dx \) using FTC!
Area Mastery Worksheet Area Mastery
Definite Integral Practice
Name:
Instructions: Use the Evaluation Theorem \( \int_{a}^{b} f(x) dx = F(b) - F(a) \) to solve each definite integral. Show your antiderivative in brackets and your substitution steps clearly.
01
\[ \int_{1}^{2} (x^3 - 2x) \, dx \]
02
\[ \int_{0}^{\pi/2} \cos(x) \, dx \]
03
\[ \int_{1}^{4} \frac{1}{\sqrt{x}} \, dx \]
Hint: Rewrite as a power.
04
\[ \int_{0}^{1} (e^x + 5) \, dx \]
Blueprint Interpretation
An engineer is designing a curved ramp. The height of the ramp is modeled by the function \( f(x) = \frac{1}{2}x^2 \) from \( x=0 \) to \( x=4 \) meters.
A. Write and evaluate the definite integral for the area under this ramp.
B. If each square meter of concrete costs $15, what is the total cost for the side of the ramp?
Evaluation Check Exit Ticket Evaluation Check
Exit Ticket | FTC Part 2
Name:
1. Evaluate the definite integral exactly:
\[ \int_{1}^{3} (4x^3 - 6x^2) \, dx \]
2. Critical Thinking:
A student evaluates \( \int_{-1}^{1} \frac{1}{x^2} dx \) and gets \( [-x^{-1}]_{-1}^{1} = -1 - (1) = -2 \). Explain why this answer is logically impossible without doing any further calculation. (Hint: Look at the graph of the function).
Movement Map Slides Movement Map
Net Change vs. Total Area
Distance vs. Displacement
The Definite Integral
\[ \int_{a}^{b} f(x) \, dx \]
Calculates Net Signed Area (Net Change). Negative area cancels positive area.
The Absolute Integral
\[ \int_{a}^{b} |f(x)| \, dx \]
Calculates Total Area (Total Distance). Every move counts as positive.
The Round Trip Paradox
"If you drive to the store and back home, your displacement is 0, but your odometer shows you traveled 10 miles."
0
Net Change
10
Total Distance
The Splitting Protocol
Three Steps to Total Area:
1 Find the Roots: Set \( f(x) = 0 \) to find where the function crosses the x-axis.
2 Split the Integral: Use the roots to break the integral into sub-intervals.
3 Absolute Value: Take the absolute value of each piece and add them together.
\[ \text{Total Area} = |\int_{a}^{c} f| + |\int_{c}^{b} f| \]
"We treat every piece of area as 'above the axis' by force."
Example: \( f(x) = x^2 - 4 \) on \( [0, 3] \)
Root: x=2
1. Net Change:
\[ \int_{0}^{3} (x^2 - 4) dx = [x^3/3 - 4x]_0^3 = (9 - 12) - (0) = -3 \]
2. Total Area:
\[ |\int_{0}^{2} f| + |\int_{2}^{3} f| = |-16/3| + |7/3| = 23/3 \]
Physics Connection
Displacement
Integral of Velocity
\[ \int_{t_1}^{t_2} v(t) \, dt \]
Distance
Integral of Speed
\[ \int_{t_1}^{t_2} |v(t)| \, dt \]
Check your worksheet for the "U-Turn Challenge"!
Distance vs Displacement Worksheet Distance vs Displacement
Net Change Application
Name:
The Splitting Technique
To find Total Distance , we must integrate speed: \( \int |v(t)| dt \). This requires splitting the integral at every point where velocity \( v(t) \) changes sign (the roots).
\( \int v(t) dt \rightarrow \) Displacement
1
The Sprints
A particle moves along a line with velocity \( v(t) = 3t - 6 \) meters per second. Find the particle's movement from \( t = 0 \) to \( t = 4 \) seconds.
A. Find the Net Displacement:
B. Find the Total Distance Traveled:
2
The Pendulum
A drone's velocity is given by \( v(t) = 10\cos(t) \) from \( t = 0 \) to \( t = \pi \) seconds.
A. Net Change in Position:
B. Total Distance Covered:
3
The Gravity Test
An object is thrown upward from a 100m cliff with an initial velocity of 20 m/s. Its velocity is \( v(t) = -10t + 20 \). Calculate the total distance it travels in the first 5 seconds.
Work Step-by-Step:
Step 1: Find when the object reaches its peak (\( v(t) = 0 \)).
Step 2: Set up the split integral for Total Distance.
Step 3: Evaluate and calculate final value.
Motion Misconceptions Guide Motion Misconceptions Guide
Lesson 4: Managing Signed Areas in Context
Priority Concepts
Definite integrals = Net Area
Absolute Value integrals = Total Area
Algebraic roots as "switching points"
The "Splitting" Visual
"Always sketch the function before finding total area. If it goes below the x-axis, you MUST split."
The Odometer Question
Ask: "If I run 5 miles North and 5 miles South, what does my GPS say? What does my fitbit/watch say?"
GPS = Net Change: Displacement is 0.
Fitbit = Total Area: Distance is 10 miles.
Fatal Algebraic Error
Students often try to calculate \( |\int_{a}^{b} f(x) dx| \) instead of \( \int_{a}^{b} |f(x)| dx \).
Explain the difference:
The first one takes the net result and makes it positive (Still net change, just absolute).
The second one makes every infinitesimal slice positive before adding them (Total Area).
Facilitating Problem 3 (Gravity)
The "Cliff" problem is the ultimate test. Many students will give a displacement of -25m or something similar.
The Wrong Way
Evaluating \( \int_{0}^{5} (-10t + 20) dt = -25 \).
The Right Way
Roots at \( t=2 \).
Differentiation Strategies
Struggling: Let them use a calculator to find the roots but require the manual setup of the two separate integrals.
Advanced: Ask them to find the Average Speed on the interval once they find total distance (Average Speed = Total Distance / Total Time).
Synthesis Slides The Synthesis
Applying the Fundamental Theorem
Pushing the Boundaries
The FTC isn't just for smooth curves. Today we apply it to:
Piecewise Functions
Absolute Value Functions
Accumulation Comparisons
The Golden Rule:
"If a function changes its definition, split the integral at the transition point."
\[ \int_{a}^{c} f_1(x) dx + \int_{c}^{b} f_2(x) dx \]
Mastering Absolute Value
How do we solve \( \int_{0}^{3} |2x - 4| \, dx \) algebraically?
1. Split Point
Set \( 2x - 4 = 0 \).x = 2 .
2. Check Signs
For \( x < 2 \), \( 2x-4 \) is negative.
3. Set Up
\[ \int_{0}^{2} -(2x-4)dx + \int_{2}^{3} (2x-4)dx \]
"Remember: The absolute value sign is just a fancy way of telling you to flip the negative parts up!"
Integrating a Piecewise
Given:
\[ f(x) = \begin{cases} x^2 & x < 1 \\ 2 - x & x \ge 1 \end{cases} \]
Evaluate \( \int_{0}^{2} f(x) \, dx \)
\[ \int_{0}^{1} x^2 \, dx + \int_{1}^{2} (2 - x) \, dx \]
\[ [x^3/3]_0^1 + [2x - x^2/2]_1^2 \]
\[ 1/3 + (4-2) - (2-1/2) = 1/3 + 1/2 = 5/6 \]
Calculus Relay
Teams of 3. Three problems. One goal.
1
The Base
2
The Split
3
The Synthesis
Wait for your team packet to begin.
Calculus Relay Challenge Activity Challenge Event
Calculus Relay
Integrating the Fundamental Theorem
Team Number
Team Score
Final Rank
01
Leg 1: The Inverse Power
Differentiate the following accumulation function. You must find \( g'(x) \) and then evaluate \( g'(2) \).
\[ g(x) = \int_{x^2}^{5} e^t dt \]
Work Area (Student 1)
02
Leg 2: The Absolute Value Split
Evaluate the total area exactly. Show the split points clearly.
\[ \int_{0}^{4} |x^2 - 4| \, dx \]
Work Area (Student 2)
03
Leg 3: The Master Synthesis
A water tank drains at a rate of \( r(t) = 4 - 2t \) liters per minute. If the tank starts with 50 liters, find the amount of water in the tank at \( t = 3 \).
Think Carefully:
Is this Net Change or Total Area?
Initial Value + Integral = Final Value?
Work Area (Student 3)
FTC Final Review Sheet FTC Final Review
Fundamental Theorem of Calculus Master Study Guide
Part 1: The Link
\[ \frac{d}{dx} \int_{a}^{u(x)} f(t) dt = f(u(x)) \cdot u'(x) \]
Use when: You are asked to differentiate an integral function.Pro-Tip: Don't actually integrate. Just "plug and multiply".
Part 2: The Shortcut
\[ \int_{a}^{b} f(x) dx = F(b) - F(a) \]
Use when: You need the exact numerical area under a curve.Pro-Tip: Use brackets to organize your work: \( [F(x)]_a^b \).
Final Mastery Tasks
1. Evaluate the absolute value integral algebraically:
\[ \int_{-1}^{3} |x - 1| \, dx \]
2. Solve the piecewise integral:
Given \( f(x) = \begin{cases} 4 & x \le 2 \\ 2x & x > 2 \end{cases} \), evaluate \( \int_{0}^{4} f(x) \, dx \).
3. Reflection Question:
We have seen that integration and differentiation are inverse operations. Does this mean that \( \int f'(x) dx \) always equals \( f(x) \)? Why or why not? (Hint: Think about the constant \( C \)).
