Name:__________________________________________ Date:_______
Accumulating Points on the AP Calculus Exam!
Objectives: By the end of the presentation, students should be able to…
1.) Understand the two parts of the Fundamental Theorem of Calculus (FTC)
2.) Recognize an integral as an accumulation of change
3.) Write an “accumulation function” and use it to solve problems
4.) Solve AP Problems involving accumulation.
Discovering the Fundamental Theorem of Calculus
The two graphs below represent the functions 𝑓(𝑥) = 𝑥
2 and 𝑔(𝑥) = 2𝑥
. 𝑓(𝑥) = 𝑥
2 y 𝑔(𝑥) = 2𝑥
. y
















    x

   






Complete the table below using the two graphs above (I’ll guide you): 𝑥
1 𝑥
2 𝑓(𝑥
1
) 𝑓(𝑥
2
0
0
1
2
0 1
) 𝑓(𝑥
2
) − 𝑓(𝑥
1 – 0 = 1
1


) Area formed by 𝑔(𝑥)
and x-axis from 𝑥
A =
− 𝑥
1
1
1
2
to 𝑥
2
.
(1)(2) = 1 1
2 bh = 1
2
( 𝑥
2
) 𝑔(𝑥
2
) =
0 3
Now continue the same process, but using these rules: i.) Areas below the x-axis are negative ii.) Areas going “backwards” are negative iii.) Not all of these will be triangles 𝑥
1 𝑥
2 𝑓(𝑥
1
) 𝑓(𝑥
2
1
1
2
3
) 𝑓(𝑥
2
) − 𝑓(𝑥
1
)
Area formed by 𝑔(𝑥)
and x-axis from 𝑥
1
to 𝑥
2
.
-1 1
-1 2
-2 1
1 -2 x


The Fundamental Theorem of Calculus – Part 1
The activity above leads us to the following conclusion:
Given
𝑏
𝑎
Or, to restate verbally: “if 𝑓(𝑥)
is the derivative of
𝐹(𝑥)
, then the integral of 𝑓(𝑥)
(area between 𝑓(𝑥) and the x-axis) from a to b is equal to the change in the function
𝐹(𝑥)
from a to b.”
For many people, this makes more sense when you bring in real-world quantities. For example:
Since velocity is the derivative of position, the integral of velocity from a to b is equal to the change in position from a to b.
Accumulation
One way to think about integrals is as an accumulation of a quantity. For example: a.) The integral of velocity is equal to an accumulation of little changes in position. b.) If S(x) is the rate at which sand is deposited on a beach (cubic feet of sand per hour), then the integral of S(x) represents an accumulation of sand. c.) If L(x) is the rate at which people enter an amusement park (people per hour), then the integral of L(x) represents an accumulation of people in the amusement park.
Based on this, we can rewrite the Fundamental Theorem of Calculus to find an accumulation: 𝑏
∫ 𝑓(𝑥)𝑑𝑥 = 𝐹(𝑏) − 𝐹(𝑎) 𝑎 𝑏
∫ 𝑓(𝑥)𝑑𝑥 + 𝐹(𝑎) = 𝐹(𝑏) − 𝐹(𝑎) + 𝐹(𝑎) 𝑎 𝑏
𝐹(𝑏) = 𝐹(𝑎) + ∫ 𝑓(𝑥)𝑑𝑥
Final value Initial Value 𝑎
= +
Change in Value (accumulation)
Here’s an example. Write an “accumulation function” to answer the question.

Typical AP Free Response “Accumulation” Problems:
Practice – Try This Problem with your Group:

AP Free Response “Accumulation” Problems – Graphical Approach:
Practice – Try This Problem with your Group:

The Fundamental Theorem of Calculus – Part 2
Picking up where we left off with “part 1”, we can use the FTC pt. 1 to write an “accumulation function” that gives the value of an accumulation for any input of x. 𝑥
𝐹(𝑥) = 𝐹(𝑎) + ∫ 𝑓(𝑡)𝑑𝑡 𝑎
Let’s take the derivative of both sides: 𝑥
𝐹′(𝑥) = [𝐹(𝑎)]′ + [∫ 𝑓(𝑡)𝑑𝑡 ] ′ 𝑎
We defined
𝐹′(𝑥) = 𝑓(𝑥)
at the beginning, so 𝑓(𝑥) = 𝑑 𝑥
∫ 𝑓(𝑡)𝑑𝑡 𝑑𝑥 𝑎
The conclusion that we can draw here is really awesome. Basically, it means that derivatives and integrals are inverses of each other, or that one cancels out the other.
Example: 𝑑 𝑑𝑥 0 𝑥
∫ sin(𝑡) 𝑑𝑡 =
Accumulation AP Multiple Choice Questions