Rizzi – Calc BC. Integrals represent an accumulated rate of change over an interval The gorilla...

4.4 THE FUNDAMENTAL THEOREM OF

CALCULUSRizzi – Calc BC

Recall…

Integrals represent an accumulated rate of change over an interval

The gorilla started at 150 metersThe accumulated rate of change was 55 meters Final position was 95 meters

In other words:

The Fundamental Theorem of Calculus

We can write this in another way:

The fundamental theorem of calculus looks at accumulated rates of change:

FTC Practice

Evaluate the integral:

FTC Practice

Evaluate the integral:

FTC Graphically

Recall the MVT

What did the MVT tell us?

How is it represented graphically?

Mean Value Theorem for Integrals

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.

MVT for Integrals Practice

Find the value of c guaranteed by the Mean Value Theorem for Integrals over the given interval

4.4 THE SECOND FUNDAMENTAL THEOREM

OF CALCULUSRizzi – Calc BC

Review: MVT for Integrals

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.

Average Value of a Function

You will NEED this for the AP exam Average value determines the average y-

value for a functionAverage Value Formula:

MVT:

Practice

Find the average value of on the interval [1, 4].

Average Value Formula:

Second FTC

The derivative of the integral of f(x) is f(x)

But why?

Second FTC Practice

Find F’(x)

More Second FTC Practice

But what about this?

Net Change Theorem

Essentially the same as FTC #1

Net Change Theorem Practice

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

Particle Motion Revisited

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)|.

Net Change Theorem Applied

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

Practice Particle Motion Problem

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?