4.3 Riemann Sums and Definite Integrals

4.3 Riemann Sums and Definite Integrals

The Definite Integral: Notation for the definite integral…

◦ b is upper limit of integration◦ a is lower limit of integration

This equals the area under a curve, above the x-axis if…

1. f(x) is continuous2. f(x) is nonnegative

( ) b

a

f x dx

Area Under a Curve

( ) when f(x) 0b

a

Area f x dx

( ) when f(x) 0b

a

Area f x dx

( ) = (area above x-axis) - (area below x-axis)b

a

f x dx

Net Area:

Thm: The integral of a constant

If ( ) on [a,b], then

= c(b-a)b

a

f x c

c dx

Integrals on the TI-84

Syntax: To evaluate …

( ) b

a

f x dx

MATH 9 fnInt(

Properties of Definite Integrals:

( ) ( ) a b

b a

f x dx f x dx 1. Order of Integration

Properties of Definite Integrals:

( ) = 0a

a

f x dx

2. Zero

Properties of Definite Integrals:

( ) ( ) b b

a a

k f x dx k f x dx

3. Constant Multiple

Properties of Definite Integrals:

[ ( ) ( )] ( ) ( ) b b b

a a a

f x g x dx f x dx g x dx

4. Sum and Difference

Properties of Definite Integrals:

( ) + ( ) ( ) b c c

a b a

f x dx f x dx f x dx

5. Additivity

Properties of Definite Integrals:

( ) ( ) a b

b a

f x dx f x dx 1. Order of Integration

( ) = 0a

a

f x dx2. Zero

( ) ( ) b b

a a

k f x dx k f x dx 3. Constant Multiple

[ ( ) ( )] ( ) ( ) b b b

a a a

f x g x dx f x dx g x dx4. Sum and Difference

( ) + ( ) ( ) b c c

a b a

f x dx f x dx f x dx 5. Additivity