Post on 18-Dec-2014
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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