Consider . It has an antiderivative , that is . On , the definite integral can be approximated as precise as one desired by Riemann sum where Since , by linear approximation

Lecture9| 1 3.3. The Fundamental Theorem of Calculus

Consider . It has an antiderivative

, that is .

On , the definite integral can be approximated as precise as one desired by Riemann sum

where

Since , by linear approximation

Lecture9| 2 This means can be approximated by

This suggests

1^st Fundamental Theorem of Calculus

If is a continuous function on and , i.e. is an antiderivative of , then

Lecture9| 3 Note It is often denoted

the bracket means subtracting the substitution by and respectively.

One does not need to put in antiderivative.

Lecture9| 4 EXAMPLE. Evaluate the following definite

integrals

Lecture9| 5 EXAMPLE. Evaluate the integral

Lecture9| 6 EXAMPLE. Find the following definite integrals

Lecture9| 7 EXAMPLE. Find the area as shown

Lecture9| 8 Some properties of definite integrals

1.

2.

3. (Additive interval property)

Lecture9| 9 EXAMPLE. Evaluate the following integrals

Lecture9| 10 EXAMPLE. Using the additive interval property, evaluate the integral

Lecture9| 11 EXAMPLE. Let

Evaluate the integral .

Lecture9| 12 4.

5.

Lecture9| 13 EXAMPLE. If

evaluate

Lecture9| 14 Net Change Theorem

The first FTC is the same as

The integral is called the net change of from to :

Lecture9| 15 EXAMPLE. A chemical flows into a storage tank at a rate of liters per minutes, where is . Find the amount of the chemical that flows into the tank during the first 20 min.

Lecture9| 16 We explain the second part of the fundamental theorem of calculus.

We shall use integration to construct functions!

Definite integral

Definite integral as functions

Lecture9| 17 Let . Consider the integral

Observe that

Another example, let . Then

and

Lecture9| 18 This is true in general.

2^nd Fundamental Theorem of Calculus

If is continuous on an open interval having ,

for any in the open interval.

Formulas with the chain rule ( numbers)

Lecture9| 19

Proof. Let . Then

For small, the integral is just one slice of the Riemann sum, so

This gives

Take the limit ,

Lecture9| 20 EXAMPLE. Evaluate the derivative

Note There is no explicit formula for

Lecture9| 21 EXAMPLE. Find the derivative of the function

Lecture9| 22 EXAMPLE. Evaluate