9. Definite Integrals

Calculus 2



Weiqi Zhou

Motivating Example

area under a curve

Settings

  • given a continuous function on

  • insert nodes into

  • take an arbitrary point
    and let

  • consider the sum

The Definite Integral

if has a limit that is independent of choices of as and ，then is called the definite integral (Riemann Integral) of on , and is denoted by

Exercise

  • take ，with

  • let ， where ，and

  • compute for and respectively

  • compute the limit of respectively and show that it is independent of the choice of

Existence

definite intgerals always exist for continuous or piecewise continuous functions on bounded and closed intervals

The Area

the signed area surrounded by the curve and the axis is defined as the value of the corresponding definite integral, and we shall agree

Exercise

a particle moves along a straight line toward a fixed direction.

its speed is at time .

how far has it traveled from time to time ?

what is the expression for its acceleration speed?

Linearity

Summability

Order Preserving

Boundedness

An Important Inequality

The Mean Value Theorem

if is continuous on ，then there exists such that

Exercises

  • compare values of and

  • give upper and lower bounds of

  • give upper and lower bounds of

Summary

  • Definite Integrals

  • Notations and Geometric Implications

  • Properties of Definite Integrals