Rizzi – Calc BC



Integrals represent an accumulated rate of
change over an interval
The gorilla started at 150 meters
The accumulated rate of change was 55 meters
Final position was 95 meters
In other words:
𝒙 𝒂 = 𝟏𝟓𝟎
𝒃
𝒗 𝒕 𝒅𝒕 = −𝟓𝟓
𝒂
𝒙 𝒃 = 𝟗𝟓

We can write this in another way:
𝒔 𝒂 = 𝟏𝟓𝟎
𝑭 𝒂 = 𝟏𝟓𝟎

𝒃
𝒗 𝒕 𝒅𝒕 = 𝟓𝟓
𝒔 𝒃 = 𝟗𝟓
𝒇 𝒙 𝒅𝒙 = 𝟓𝟓
𝑭 𝒃 = 𝟗𝟓
𝒂
𝒃
𝒂
The fundamental theorem of calculus looks at
accumulated rates of change:
𝒃
𝑭 𝒂 +
𝒇 𝒙 𝒅𝒙 = 𝑭(𝒃)
𝒂
𝒃
𝒇 𝒙 𝒅𝒙 = 𝑭 𝒃 − 𝑭(𝒂)
𝒂

Evaluate the integral:
2
(𝑥 2 − 3)𝑑𝑥
1
2
−
3

Evaluate the integral:
4
3 𝑥𝑑𝑥
1
14

What did the MVT tell us?

How is it represented graphically?

The MVT for Integrals says: somewhere in the
interval [a, b] there is a f(c) value that accurately
approximates the area of the curve under the
interval.
𝑏
𝑓 𝑥 𝑑𝑥 = 𝑓(𝑐)(𝑏 − 𝑎)
𝑎

Find the value of c guaranteed by the Mean
Value Theorem for Integrals over the given
interval
𝑥2
𝑦 = , [0, 6]
4
𝑐 =2 3
Rizzi – Calc BC

The MVT for Integrals says: somewhere in the
interval [a, b] there is a f(c) value that accurately
approximates the area of the curve under the
interval.
𝑏
𝑓 𝑥 𝑑𝑥 = 𝑓(𝑐)(𝑏 − 𝑎)
𝑎


You will NEED this for the AP exam
Average value determines the average y-value
for a function
Average Value Formula:
𝑏
1
𝑓 𝑥 𝑑𝑥 = 𝑓(𝑐)
𝑏−𝑎 𝑎
𝑏
MVT:
𝑓 𝑥 𝑑𝑥 = 𝑓(𝑐)(𝑏 − 𝑎)
𝑎

Find the average value of 𝑓 𝑥 = 3𝑥 2 − 2𝑥 on
the interval [1, 4].
Average Value Formula:
𝑏
1
𝑓 𝑥 𝑑𝑥 = 𝑓(𝑐)
𝑏−𝑎 𝑎

The derivative of the integral of f(x) is f(x)

But why?

Find F’(x)
𝑥
𝐹 𝑥 =
1
2
𝑡
𝑑𝑡
2
𝑡 +1

But what about this?
𝑥3
𝐹 𝑥 =
𝜋
cos 𝑡 𝑑𝑡
2

Essentially the same as FTC #1

A chemical flows into a storage tank at a rate of
180 + 3t liters per minute, where 0 ≤ t ≤ 60. Find
the amount of the chemical that flows into the
tank during the first 20 minutes.
4200 liters
When calculating the total distance traveled by
the particle, consider the intervals where v(t) ≤ 0
and the intervals where v(t) ≥ 0.
When v(t) ≤ 0, the particle moves to the left, and
when v(t) ≥ 0, the particle moves to the right.
To calculate the total distance traveled, integrate
the absolute value of velocity |v(t)|.
So, the displacement of a particle
and the total distance traveled by
a particle over [a, b] is
and the total distance traveled by the particle on [a, b] is
The velocity (in feet per second) of a particle
moving along a line is
v(t) = t3 – 10t2 + 29t – 20
where t is the time in seconds.
a. What is the displacement of the particle on the
time interval 1 ≤ t ≤ 5?
b. What is the total distance traveled by the
particle on the time interval 1 ≤ t ≤ 5?