Review 5'1 Estimating with Finite Sums

1
Review 5.1 Estimating with Finite Sums

• Area
• To estimate the area of the region R under the
  graph, two different rectangles are used
  (inscribed rectangles and circumscribed
  rectangles)

2

• The sum of areas of circumscribed rectangles is
  called an upper sum.
• The sum of areas of inscribed rectangles is
  called a lower sum.

3
Summary

• By taking more and more rectangles, with each
  rectangle thinner than before, the finite sums
  give a better and better approximation to the
  true area of the region R.

4
5.2 Sigma Notation and Limits of Finite Sums

• Finite Sums and Sigma Notation
• Sigma Notation
• The sum of n terms is written as
• The Greek letter stands for sum. The index
  of summation k tells where the sum begins and
  where the sum ends

5
Examples

• 2) Examples
• (a)
• (b)

6
Algebra Rules

• 3) Algebra Rules
• Sum rule
• Difference Rule
• Constant Multiple Rule
• Constant Value Rule

7
Some Formulas

• a)
• b)
• c)

8
Examples

• a)
• b)

9
Limits of Finite Sum

• 2. Limits of Finite Sum
• By taking more and more rectangles, with each
  rectangle thinner than before, the finite sums
  (both lower sum and upper sum) give a better and
  better approximation to the true area of the
  region R. So the limit of the finite sum is the
  true area.

10
Riemann Sums

• 4. Riemann Sums
• We begin with any function f defined on
• a, b. Now divide the interval a, b into n
  subintervals.
• a, x1, x1, x2, x2, x3, ., xn-1, b
• The set Pa, x1, x2,.., b
• x0, x1, x2,.., xn
• is called a partition of a, b

11

• In each subinterval xk-1, xk we select a point
  ck. Then the area of the small rectangle over
  xk-1, xk with height f(ck) is
• f(ck)(xk- xk-1) f(ck) ? xk
• We sum all these areas to get
• This sum is called a Riemann sum for f on the
  interval a, b. The largest width of all
  subintervals is called the norm of the partition
  P and is denoted by P.

12
5.3 The Definite Integral

• Limits of Riemann Sums
• Definition
• Let f be a function defined on a, b. If the
  limit of the Riemann sum is
  I, then I is called the definite integral of f
  over a, b. And we say that f is integrable over
  a, b.
• 2) Notation

13
Theorem

• 3) Existence of the Definite Integral
• A continuous function is integrable. That is, if
  a function f is continuous on a, b, then its
  definite integral over a, b exists.

14
Properties of Definite Integrals

• 4) Let f(x) and g(x) be integrable funtions on
  a, b. Then
• a)
• b)
• c)
• d)

15
Area Under the Graph of a Nonnegative Function

• 2. Area Under a Curve as a definite Integral
• If f(x) is nonnegative and integrable on a, b,
  then the area under the curve of f(x) over a, b
  is the definite integral of f from a to b.

16
Example

• Ex.1 Compute
• Solution

17
Example
18
Example

• Ex.1 Compute
• Solution

19
Homework

• 1-11 odd, 15, 17, 19, 23 on page 342-343.
• 1-13 odd, 27, 29, 39, 43 on page 352-353.
• Exam 3 Wednesday (11/8)
• Coverage 4.2-4.5, 4.8, 5.1-5.3