Definite Integral and

the Fundamental Theorem of Calculus

Definition of Definite Integral If f is defined on the closed interval [a, b] and the limit of Riemann sums over the partitions ᅀ exists,

then f is said to be integrable on [a, b] and the limit denoted by

The limit is called the definite integral of f from a to b. The number a is the lower limit of integration and the number b is the upper limit of integration.

Properties of Definite Integrals

EXAMPLE 1

4I I et et

IT 4

21T I 3 211

5 1251a

**RECORD THIS IN YOUR PINK SHEET!

Example 1

taketheantiderivative 2keepthelimitsofintegration IgX3 3 x

I

Evaluatetheantigeminmeyngaumoenniang 213 312 Ill 3111hmOfCSabfaddx Fla Fla 8

3 6 Ig f 3 Y y2 3

andthen asimplify

Iz 3 Iz za µs

Iz IEEE.eesnretsneneEotTeaana9ne

sneyeainIs ii

3 theyaxis

Example 2 tieRewriteso

iiiinertia 3fix'tDXtaketheMTderivativesina.si it 3 fExZ 2X

vmaideariemmeiat 21452 2 1 3andaccordingto thendamental1hmCalabfaddxFcbFca I 6 2simplify 4

Example 3