The Fundamental Theorem of Calculus ...
Who needs Pythagoras' theorem

A particle is traveling along a straight line. Its position is given by
Find the change in position from t = 1 to t = 4.

Suppose a car is moving with non-decreasing speed according to the table below:
(a) What is an upper estimate for the distance traveled in the first 2 seconds?
(b) Determine upper and lower estimates for the change in position for
the first 10 seconds.

The Fundamental Theorem of Calculus ... part I
Let the function ƒ be continuous on [a, b] with derivative ƒ'. Then ...
Total change in ƒ
In words: Integrating a rate of change function ƒ' over an interval [a, b]
gives the total change in ƒ, ƒ(b) - ƒ(a), over the same interval.
In other words, the integral of a derivative is the same thing as
the total change in it's parent function over the same interval.
Also recall:
and ...

t hours after a storage tank develops a leak, oil is spilling onto the ground at a
rate given by: gallons/hour. How many gallons have spilled out
during the first three hours?

Question:
change in ?????

If Then Determine
(a) exactly.
(b) approximately, using fnInt.

If Then Determine
(a) approximately, using fnInt.
(b) exactly.